







(2018^2019)(2019^2018) 
20190123 

From Nazrul: Find the remainder when (2018^2019)(2019^2018) is divided by 4? Please show me the process. Answered by Harley Weston. 





4 digit "combinations" 
20180518 

From Marvin: I am designing a game, and I need away of figuring this out without having to write numerically all these numbers
4 digit combinations 0 through 13 nonrepeating Answered by Penny Nom. 





What is small fourdigit whole number? 
20180415 

From Rajesh: What is small fourdigit whole number? Answered by Penny Nom. 





Base 6 to base 4 
20180325 

From Malik: Hi Sir..!
I just want to know the proper example with a brief explanation of conversion of base 6 into base 4.
kindly reply me as soon as possible..
Thank you Answered by Penny Nom. 





A four digit pass code 
20180226 

From Jessica: I was trying to fond out how many combinations could there be for a four digit pass code to a phone when no numbers are repeated? Answered by Penny Nom. 





4444^4444 
20171210 

From Sashi: 4444 to power of 4444=?
Please share the result with simplification. Answered by Penny Nom. 





Sampling distribution 
20171016 

From Esther: I need 4digits combination of 235689 and without replacement. Thanks Answered by Penny Nom. 





A 4 sided lot 
20170815 

From Micheal: If one side of .4 acre is 25 feet what would the other three sides be Answered by Penny Nom. 





Teams of 4 from 20 people 
20170804 

From Christopher: math team has 20 people on the team but only 4 can enter the competition. The team captain has to pick 4 of them but he knows that he cannot put James, Bill or Todd on the team with one another or there will be problems. what would the formula be to find the solution to how many possible team combinations the are Answered by Penny Nom. 





A 4 digit safe combination 
20170729 

From Donna: I lost my safe combination :( it is 4 digit # with the last digit being 2.
I do not think the 1st digit is 1, 2 or 3
Can u help? Thank u :) Answered by Penny Nom. 





Some 4 digit numbers 
20170724 

From Camille: I need all of the 4 digit combinations that starts with number 3 and digits can't be repeated Answered by Penny Nom. 





The smallest and largest 4 digit integers 
20170613 

From Avneetpal: Difference of the smallest and biggest 4digit number Answered by Penny Nom. 





4 digit codes with repeating digits 
20170601 

From Morgan: what are all possible 4 digit code repeating numbers? cuz i know that you have one website on it but, it doesn't repeat 2 numbers in a code. Answered by Harley. 





A 40 acre square 
20170514 

From Richard: How many feet on each side of 40 acre square ? Answered by Penny Nom. 





Is a large integer divisible by 2^n? 
20170328 

From Sahand: You are given number x that is very large (that large that it can't be divided by hand)
can we find out that x is divisible by 2^n or not? Answered by Penny Nom. 





More on calculating the area (acreage) of a foursided lot 
20170306 

From Kerri:
Can you please show the formula used to calculate areas as provided on this page:
Calculating the area (acreage) of a foursided lot Answered by Penny Nom. 





Four digit numbers 
20160908 

From ning: 00009999 there is ten thousand 4digit numbers. How to find the number of ways that 4 digit are the same? How to prove it by using formula? Answered by Penny Nom. 





A 1.4 acre square 
20160628 

From Jamie: Dear Faithfullness,
I was previously to be purchasing a square 1 acre plot, which I believe was to have 209feet, 4 sides perimeter.
How big would a 1.4 acre site be?
In sq feet length ie original 209ft + (the adeed 0.4) Answered by Penny Nom. 





The area of a 4sided lot 
20160525 

From prasad: How to find area of a land whose sides are 41ft,33ft,32.3ft and 33.2 ft. Pl give me the formula and proof. Answered by Penny Nom. 





A six digit code 
20160506 

From Ralph: How many combinations are possible if a "safe" has four buttons and you can use a six digit code? (these are just single press buttons..)For example, this could be one possible combination: Button 4 Button 1 Button 1 Button 3 Button 2 Button 4. Thanks! Answered by Penny Nom. 





How can I express Sin 4A in term of Sin A? 
20160222 

From Odukoya: How can I express Sin 4A in term of Sin A? Answered by Penny Nom. 





Subtraction in a base larger than ten 
20160131 

From Jenalee: Hello,
So happy I found this site. I am having problems with bases. My question is 97B9  6A3A base 16 for both.
Please help. Answered by Penny Nom. 





Four consecutive integers 
20160105 

From sherica: my teacher gave us an assignment and it says here that;
THERE ARE FOUR CONSECUTIVE INTEGERS.IF THE SMALLEST ONE IS x ,FIND THE SUM OF THE FOUR INTEGERS. that is the question ...thanks hope that u can answer it Answered by Penny Nom. 





The perimeter of a 4 acre square 
20151216 

From samantha: What is the distance in miles or how many laps to make a mile, if one was walking around 4 acres of land which is like a square? Answered by Penny Nom. 





A 4 by 7 array 
20151027 

From star: don breaks a 4x7 array into a 2x7 array and another array.what is the fact for don's second array Answered by Penny Nom. 





A fantasy football league 
20150611 

From Jason: Hello. We have a fantasy football league with 14 teams. The season is 13 weeks long and we would like to figure out a way to have two divisions of seven teams each with each team playing their interdivision opponents twice. Is there a way to set this up? I cannot seem to find a way to do it. Thanks! Answered by Victoria West. 





Golf for 14: 2 fourballs and 2 threeballs 
20150520 

From martin: I have to organize 14 players to play four rounds of golf. Each round comprise two fourballs and two three balls .
What's the best combination to ensure that each persons plays with as many of the others as possible without playing with somebody more than twice? Answered by Victoria West. 





The game of 24 
20150328 

From Suzanne: I love this game! Cannot figure out 4,9,10,16 to equal 24? Help! Answered by Penny Nom and Claude Tardif. 





The game of 24 
20150121 

From aaron: 24 game whants me to use 11 14 5 and 9 to make 24 allowed (+  dividing and multiplying) Answered by Penny Nom. 





4 card hands 
20141102 

From Ronaldo: How many Mus hands (combination of 4 cards allowing repetitions)
can be made from a deck containing 8 types of cards?
RRRR, RRRC, RRRS, RRR7, RRR6, RRR5, RRR4, RRRA,
RRCC, RRCS, RRC7, RRC^, RRC5, RRC4, RRCA,
RRSS, RRS7, etc.
I think 330. If so, what is the formula? Answered by Penny Nom. 





logbase4 (sin x) = 0.5 
20141026 

From Susan: Solve for x: logbase4 (sin x) = 0.5 Answered by Penny Nom. 





A 14 team golf league 
20140711 

From Pat: We have a 14 team golf league that plays a 16 week schedule. Playing once per week, I need a schedule where each team plays every other team at least one time. Answered by Victoria West. 





A rotation schedule for 14 teams 
20140708 

From Daniel: I am organizing team activities for a summer camp. We have 14 teams and 7 rotations and 7 different games. every team will play a different game each rotation. Where I am having trouble is I would like each team to play a different team each rotation without a repeat. Is that possible? if so what is the formula? Answered by Victoria West. 





14 golfers, 7 men and 7 women 
20140706 

From John: I have to schedule 14 golfers, 7 men and 7 women, into 4 teams each day, playing for 3 days. So each player plays with as many different players as possible for 3 games. Answered by Victoria West. 





How can eight eights be 1000? 
20140620 

From Aamir: The digit 8 if we add it 8 time in it the answer should be equal to 1000.it may possible.i.e.,
8+8+8+8+8+8+8+8=1000 Answered by Penny Nom. 





14 golfers for 13 weeks 
20140517 

From Judi: I want to schedule each of my 14 golfers to play each of the other girls one time during a 13 week period. We would golf in foursomes , threesomes or twosomes to make this happen. Can you send me a grid that would do this scheduling? I'm at a loss as to how to do it!! Answered by Victoria West. 





Golf for 14 players over 12 weeks 
20140516 

From Judi: How would I develop a playing schedule using foursomes and one twosome for 14 players who will play 12 weeks and each person would play all but one of the others? Answered by Victoria West. 





4 golfers in twosomes 
20140504 

From Lori: 4 golfers want to play as 2 somes every week, but rotate the 2 players so we all play together
With each other evenly thru out the summer Answered by Victoria West. 





1÷[1√2(order of surd is 4)] 
20140502 

From Anoushka: if t=1÷[1√2(order of surd is 4)] , then t=? Answered by Penny Nom. 





A golf league consisting of 14 two man teams 
20140410 

From Kevin: Hello,
I need to make a schedule for a golf league consisting of 14 two man teams playing over the course of 20 weeks. I understand that each team will play at least once and I believe 7 teams will play each other twice over the course of the season. How would I generate this schedule. Thank you,
Kevin Perry Answered by Victoria West. 





A 4 digit number 
20140404 

From LIM: "A" is a 4 digit number formed by all the numbers from 1 to 4. When "A" is divided by 9, the remainder is the biggest possible value. What is the biggest value of A? Answered by Chris Fisher. 





A schedule for an 8 man golf team playing 4 each week 
20140401 

From Harold: We have an 8 man golf team and wish to schedule only 4 to play each week and to have each member play with one of the other 7 at least once.
What is the simplest method. Number of weeks could be up to 18 or so. Answered by Victoria West. 





12 golfers, 24 weeks 
20140327 

From George: I have 12 golfers and want to set up a season schedule that will avoid duplication of playing partners as much as possible. The saeson is 24 weeks long. Do you have a formula? Answered by Victoria West. 





14 teams divided into two equal divisions 
20140318 

From C: We have 14 teams divided into two equal divisions. We want each team to play each team in their division one time and four teams from the other division one time. What is the schedule Answered by Victoria West. 





24 golfers playing 3 rounds 
20140310 

From Jim: How can I rotate 24 golfers playing 3 rounds of golf so that none of them play with the same person twice. Answered by Victoria West. 





Four digit phone numbers 
20140225 

From Tom: Hi, I'm an online math teacher, working on lessons for my students.
This question is in their book and I can't do it...! Help!
It has to do with phone numbers, and in this problem,
we are only dealing with
the last 4 digits of the number.
It asks how many possible numbers we have
if at least one digit repeats in the last 4 digits of the number. Answered by Penny Nom. 





The graph of f(x)=x^24 
20140207 

From Allison: Graph f(x)=x^24 and give the five point of the x/y intercepts. Answered by Penny Nom. 





A 4 digit number 
20131220 

From swathi: How many 4 digit numbers can be formed using the digit 3 only once but the digit 3 must be used in all numbers? Answered by Penny Nom. 





Schedule for a 4 team basketball league 
20131208 

From Sommer: I need a schedule for my kids basketball league. There are 4 teams.
We will play every Saturday for 4 weeks with 4 games per Saturday.
Week 5 will be the playoffs where 1st will play 4th and 2nd will play 3rd.
Week 6 will be winners of week 5. Please help!! Answered by Victoria West. 





Four tangent circles 
20131009 

From Nilesh: Four circular cardboard pieces, each of radius 7cm are placed in such a way that each piece touches two other pieces. How to find the area of the space enclosed by the four pieces?
Please let me know. Answered by Robert Dawson. 





Two teams of 4 playing 4 rounds of golf 
20130929 

From Tom: We would like to play a ryder cup format with two teams of 4 playing 4 rounds of golf. Three of the rounds would be pairs playing each other and one round would be singles. How can we set up the foursomes so we balance the number of times we play with every other golfer. Answered by Victoria West. 





14 golfers in two person teams 
20130904 

From Stewart: How to rotate 14 golfers in two person teams without duplication until all have been paired at least once. Play is once per week. Answered by Victoria West. 





The perimeter of a 1.4 acre square 
20130829 

From Destini: I cannot figure out what the perimeter for 1.4 acres of land
in the shape of a square. I have tried and tried and eventually
tried to use a calculator and I still can't figure it out. Can you
help me? Thanks!
Destini Answered by Penny Nom. 





Tesseract 
20130822 

From Dan: I am not a mathematician. This seems to me an intuitively simple enough problem that I very much need an answer to from someone who's mathematics are better than mine. Please help.
The question is: for a tesseract of side length = 1
what is the distance of the center of each cube from the center of the tesseract ?
I think I have calculated the distance of each vertex from the center, and of the center of each edge from the center, but the question above baffles me.
(anyone not having a clue what I am talking about can brush up here http://en.wikipedia.org/wiki/Tesseract )
Thanks in advance  Dan V Answered by Robert Dawson. 





4 couples golfing 
20130706 

From Brian: We have 4 couples going on a 4 day golf vacation playing 4 rounds of golf.
I have spent hours trying to set up a schedule that allows the 4 spouse to play together,
and then each spouse to play with one of the other spouses (men with women) for the]
other 3 rounds.
I would like the foursomes to be different as possible. Also, noone should RIDE in a cart with the
same person more than once.
I am not a math guy so I try to do this by working it out on paper, over
and over again. It ain't working!!
If you can help, I am very thankful.
Brian Answered by Victoria West. 





4 linear equations with 3 unknowns 
20130412 

From Marian: how to solve for 3 unknowns in 4 simultaneous equations Answered by Penny Nom. 





The distance between 2 lines in 4D 
20130401 

From samane: how can i measure the distance between 2 lines in 4D?? Answered by Chris Fisher. 





A 4 digit code 
20130326 

From chris: I saw a previous question about a 4 digit code type thing and I was wondering if you could email me that same type of thing.
I changed the pin number on my phone's lock screen and I tried hard to remember it, then I had to go help my mom with something and I automatically forgot it and there is no way for me to get back on my phone.
I need all of the possible 4 digit number combinations without the numbers 0, 1, or 2. It cannot be repeating, ascending, or descending.
I know you may not want to do all of them, but I really need it. Answered by Robert Dawson. 





6 digit permutations from {149} 
20121227 

From ange: how many six digits numbers permutations can be formed with the set {149} if no digit is used twice in each permutation?? Answered by Penny Nom. 





A 4 digit phone lock using the digits 4, 2, 3 and 0 
20121224 

From Ragu: I would like to know formula for searching a combination of 4 digit. eg.
I knew my Phone Lock combination is my car plate number 4230. But
this time, i used another way around instead of 4230.
But I am sure, there is only this 4 digits only. 4 & 2 & 3 & 0 Answered by Penny Nom. 





Some 4 digit numbers 
20121025 

From samira: How many 4digit numbers are there in which the first and last digits are the same? Answered by Penny Nom. 





4 digit combinations 
20121017 

From Scott: How many different 4 number combinations from 0123456789 with 2 out of the 4 odd and 2 out of the 4 even, with no numbers repeating. Thanks Answered by p. 





A four digit number 
20121009 

From cyndi: find this 4 digit number: the tens digit is 1/3 of the hundreds digit; the 2 digit number formed by the tens and ones digits i the largest prime number smaller than 40; the sum of all 4 digits is the same as the number of face on an icosahedron ? what is the number? Answered by Penny Nom. 





Four tangent circles 
20121004 

From renu: inside of a circle K of radius length measure R,three circular discs A,Band C each of radius r are placed so that each touches the other two and K . express R in terms of r. in the space between K, A and B , another circular disc D is placed which just touches K, A and B. if the radius is s, show that (6+root3)s=(2+root3)r Answered by Penny Nom. 





A sand box with 14 sides 
20120710 

From Tom: Want to build a sand box for a pool that is fourteen foot round wanted to make it in fourteen pieces of wood what angle would each piece be and how long would each piece be Answered by Harley Weston. 





6/49 
20120525 

From Mark: Can you explain why the lotto 6/49 (a lottery of 49 numbers and 6 slots) is a combination and not a permutation?
It seems to me that it would a permutation given that the numbers are written in numeric "order" when you look them up in the newspaper.
I realize that 49P6 is orders of magnitude larger than 49C6. But I am confused about the reasoning behind it. Answered by Robert Dawson. 





Golf: 8 players over 4 days in 4balls 
20120503 

From JimB: I cannot seem to get this to work. Previous answers cover 2balls, but we are 4balls.
What we really want is for each player to play with each other, at least once and no more than twice.
Ideally, each would also play equally in the first and the second 4ball of the day. Answered by Victoria West. 





A 14 man golf league 
20120412 

From Paul: 14 man golf league. Need to have alternating 2 man teams for 18 weeks. Need to rotate foursomes so no one is stuck in twosome more than once. Can you help? Answered by Robert Dawson and Victoria West. 





240 acres 
20120217 

From jon: How many linear feet are in a 240 acre rectangle piece of property Answered by Harley Weston. 





Converting from base 4 to base 10 
20111129 

From anjg: How to convert base 4 number 2^11 to base 10 number? Answered by Penny Nom. 





4 x 2 
20110830 

From Lucjan: how to sketch a diagram that shows 4x2 expression? Answered by Penny Nom. 





58 choose 4 
20110826 

From Jean: Using only the numbers 159, how many times would the
number 23 appear in combinations of 5 numbers?
... Can a list be provided as well?
Thank you very much,
Jean Answered by Penny Nom. 





A group of 40 golfers 
20110711 

From Doug: We are a group of 40 golfers who play in 10 x foursomes playing 3 rounds. Can we all play with different players each round yet maintain the foursomes format. Answered by Victoria West. 





Golf for 14 
20110711 

From Doug: We have 14 players playing over 4 days. can we play with different players every day and keep the 3 x foursomes and 1 x pair format Answered by Victoria West. 





The game of 24 
20110506 

From shaylin: how do you use 3,9,7,9 once in any operation equal to 24? Answered by Penny Nom. 





Golf pairings 
20101217 

From AL: I have a golf tourney where we have fourteen players and are playing five times/rounds of golf. I am trying to pair up two threesomes and two foursomes so we don't play with the same person twice or the least amount of times. Answered by Victoria West. 





The game of 24 
20101026 

From gabrielle: What is the answer using the numbers 0 0 0 0 (trig and calc are
allowed, all math is allowed) in the game 24? Answered by Claude Tardif. 





The area of a 4 sided region 
20100714 

From Cliff: If I wanted to know the square footage of an area that has four different sides and only had the length of one side and the angles at the ends of that length; could the other lengths be found if I wanted to have a total of 130,680 square feet? Answered by Robert Dawson and Tyler Wood. 





Four circles in a square 
20100604 

From Daniela: four circles are drawn in a square such that the circles are tangent to each other as shown. find the area of the shaded region. It the goes on to show a diagram with a square and four circles drawn in it. The length of a side of the square is 24. Please help me! Answered by Penny Nom. 





More on carbon 14 dating 
20100526 

From Malachi: I am doing carbon 14 dating and i figured out the rule for taking the percent of the original C14 to figure out how old it is. how do I work the equation the other way around to find the percent remaining, from how long ago the objects organic mater died Answered by Harley Weston. 





A 14 side well house cover 
20100312 

From Kenneth: I am 35 yr I am wanting to build a well house cover. I'm trying to figure out how long the pieces need to be and what angle they need to be for a 4 ft dia with 14 side well house. I would love an answer but would also like to know how to figure it in the future. Thanks Kd Answered by Harley Weston. 





The game of 24 
20100220 

From haley: hello.... we have these very hard make 24 stumpers. i am stuck on one.... how can you make 24 using the numbers 771318?? thanks! Answered by Claude Tardif. 





4digit positive numbers 
20100215 

From marjon: Find the sum of all the 4digit positive numbers with no zero digit. Answered by Penny Nom. 





1^3 + 2^3 + 3^3 +4^3 ... n^3 = ? 
20100129 

From ireimaima: Hi..
Can u please help me with this question..
I find that when i test eg: n=2 for n (n+1) /4,
it seems that it does not giving me the right answer of 1^3 + 2^3 = 9
but 3/2... i'm confuse..can u please help me..thanks so much
Prove that:
1^3 + 2^3 + 3^3 +4^3………………………………..n^3 = n (n+1) /4 Answered by Penny Nom. 





A 4digit phone code 
20100121 

From pat: my daughter can`t remember her 4digit code to unlock her phone. she is pretty sure it starts with a 4. can you help us? my brain is locked up Answered by Robert Dawson. 





Expressing the natural numbers in terms of 4 fours. 
20100119 

From Steve: When I was studying mathematics, there were several of us that tried to express the natural numbers in terms of 4 fours. Allowing the use of any known math operation, for example n to the zeroth power is 1.Is there more info on this exercise. This was done mentally, nothing written down. Answered by Robert Dawson. 





Four digit numbers with no repeats 
20100102 

From Daniel: I want to know how many different numbers there would be between the numbers
00009999 that do not have any numbers that duplicate or repeat in the number? For example
the first number would have to be 0123 and the last would be 9876 (I think) Thanks! Answered by Penny Nom. 





A 4 digit number 
20091215 

From Lisa: Billy Club was assigned the task of putting numbers on
all the playground balls used during daily recess. Billy
will number the balls using the following rules:
1. It will be a 4digit number.
2. The digit in the thousands place will be a 1 or a 2.
3. The digit in the hundreds place will be a 2, 4, or 6.
4. The digit in the tens place will be an odd number.
5. The digit in the ones place will be greater than six.
How many balls can Billy number if he follows these rules? Answered by Robert Dawson. 





Game of 24 
20091209 

From Tony: Checked out all of the previous inquiries into the crazed game, and found none with these particular numbers: 1, 4, 6, 7. I've tried every method I can think of, please Help! Tks. Answered by Robert Dawson, Penny Nom and Claude tardif. 





Four consecutive odd integers 
20091024 

From Tonya: the sum of four consecutive odd intergers 2008 what are the intergers Answered by Penny Nom. 





24 golfers, 6 days 
20091003 

From patrick: i need a pairing schedule for 24 golfers for six days playing in foursomes. Please Answered by Victoria West. 





Game of 24 
20091002 

From trami: game of 24 by using 4 7 7 1? Answered by Claude Tardif. 





24 golfers 
20091001 

From Peter: I am working on a 19 week golf schedule with 24 golfers, and I would like to know how to make up six foursomes each week without having a player paired with the same golfers. Answered by Victoria West. 





Game of 24 
20090924 

From vanessa: Using the following numbers 13,6,6, and 12 to equal 24. Using 5th grade math Answered by Penny Nom and Claude Tardif. 





24 golfers: 4 days 
20090921 

From Steven: We have 24 golfers (6 A's, 6 B's, 6 C's & 6 D's) playing 4 days.
Each day a foursome consist of an A, a B, a C & a D player.
Can you give me a pairing list so that no two golfers will play on the same foursome for the 4 days? Answered by Chris Fisher and Victoria West. 





Dart teams 
20090918 

From Sally: I have 14 dart teams. Each team has a home bar.Usually it is one week home, the next week away. I have 3 locations that have one board only, yet have 2 teams at each location. I have one location that has 2 dart boards, and 3 teams. The last location have 3 dart boards and 4 teams.Can you make me a schedule that will be home and away, and every teams plays each other twice. In case you are wondering, the bars sponsor the teams.Thank you. Sally Answered by Robert Dawson. 





The game of 24 
20090827 

From ANITA: Help I'm suppose to be a math expert, yet I can not yield 24 from the numbers 8,4,9,9, using each only once. HELP Answered by Penny Nom. 





The game of 24 
20090826 

From melvi: how can you get the numbner 24 but only useing the numbers 7,5,4,5 Answered by Penny Nom. 





The smallest 4 digit number 
20090617 

From Kat: What is the smallest 4 digit number? 0001 or 1000 Answered by Robert Dawson. 





Game of 24 
20090522 

From Sam: Using the numbers 7, 3, 2, and 2, how do you get the number 24? Answered by Claude Tardif and Harley Weston. 





Combinations and permutations 
20090507 

From Renee: We are trying to figure out the equation or how to solve the problem of how many combinations of 4 digits you can get out of 09 without repeating the numbers in a set (i.e., no 0001, 0002) begining with 0123, 0124, 0125 etc
Thanks! Answered by Stephen La Rocque. 





Game of 24 
20090502 

From Barbara: using the numbers 1,3 7 and 9 only once how do you get the answer of 24 Answered by Claude Tardif. 





Fourdigit combinations 
20090418 

From Carri: How many 4digit combinations can be made out of digits 1, 2, 3, 4, & 5 using each number once? What are the combinations? Answered by Penny Nom. 





Fourdigit numbers 
20090411 

From jp: How many fourdigit numbers are there? Answered by Penny Nom. 





Game of 24 
20090408 

From Joe: How do you get 24 using 6,9,10,10 once. Its called the 24 game.
Joe Answered by Harley Weston. 





Game of 24 
20090405 

From Ramona: math 24 using 16, 9, 20, 7 Answered by Penny Nom. 





0123456789 
20090325 

From Dr.: Many years ago I discovered that there are 33 numbers which,
when multiplied by 0123456789, simply rearrange the digits. Surprisingly
(to me at least) is the fact that the digits of all of those 33 numbers add up
to one of the 6 "cardinal" numbers 1, 2, 4, 5, 7, and 8. For example
6 + 2 = 8 and 62 X 0123456789 = 7654320918 etc.
Another interesting (to me) fact is that only the cardinal numbers
1, 2, 4, 5, 7, and 8 rearrange the digits in the number 9876543210.
My question is: "Is this known to anyone but me and the people I have
told?" Answered by Robert Dawson. 





Choosing a 4 digit number at random 
20090324 

From shabkhal: If a fourdigit number is chosen at random, what is the probability that the product of the digits is 12. Answered by Harley Weston. 





4 digit numbers from 1159 
20090310 

From David: I have the digits "1159" and need to come up with a complete list of possible fourdigit combinations. It does NOT matter the order of the two number ones (1), so this will help eliminate many possibilities.
Separately, it would also help to actually know the actual number of combinations with which I am dealing to start.
David :) Answered by Penny Nom. 





Four digit numbers 
20090310 

From Theresa: I need to know the number of 4 digit numbers that can be formed using 1234 without repeating numbers Answered by Penny Nom. 





A 14 team league that is divided into 2 divisions 
20090307 

From Ronnie: I have a 14 team league that is divided into 2 divisions. We have a total of 18 weeks to play. I would like for each team to play the team in their division twice. The remaining 6 weeks would be played with the other division. Any help would be appreciated. Answered by Chris Fisher and Victoria West. 





4 digit strings starting with 7 
20090303 

From Angel: I need to no all the 4 digit combinations starting with 7 and with no repititions. Answered by Penny Nom. 





The game of 24 
20090301 

From Jeff: My son in the 5th grade was given a problem for extra credit and could use parents help. He was given the numbers 21, 13, 9, 6 and asked to use each number only once by +, , /, x the numbers in any order to obtain an answer of 24. Answered by Harley Weston. 





Buying 13,983,816 tickets for the 6/49 lotto 
20090223 

From luis: if 6/49 lotto combinations is 13,983,816 combinations at 10.pesos per combinationthe price amount is 139,838.160.00 is it wise for me to get all the combinations since the the pot money is more than 347 million pesos. Am I right? Answered by Robert Dawson and Penny Nom. 





Four digit numbers using 1, 0, 2 and 8 
20090127 

From Mike: How many combinations of this four digit number 1028 can this produce only using those numbers. Example would be 0000, 0001,0002,0008,0010,0011,0012,0018,etc... Answered by Penny Nom. 





A four digit passcode 
20090112 

From john: i have a machine at home that has a four digit passcode and the numbers range from 1 to 4 how can i get all the ways those numbers can be used Answered by Harley Weston. 





A 4digit number 
20090105 

From Anya: It is a 4digit number
The tens are double the ones.
The thousands are double the tens.
The sum of the digits is 19.
What number is it? Answered by Penny Nom. 





24 golfers playing golf for 9 rounds 
20081223 

From Duane: Got 24 golfers playing golf for 9 rounds. Any formula where everyone can play with everyone else at least once. We are playing 4somes only. Thanks, Duane Answered by Victoria West. 





Golf  24 guys 4somes 6 rounds 
20081207 

From Duane: Thank you. What about 24 guys 4somes 6 rounds everyone playing with each
other at least once.
Thanks again.
Duane Answered by Victoria West. 





The game of 24 
20081124 

From Davey: this has us all stumped. 5 ,5 ,7 ,and9 = 24. Oh may you assist in ending
our brain pain Answered by Penny Nom. 





A 4 digit number 
20081106 

From Chris: numbers 0  9 form a 4 digit number with no repetition.
It would round to 3000 if rounded to the nearest thousand
The digit in the tens place is twice the value of the digit in the thousands place.
It is an odd number.
It contains the largest digit.
It contains one digit less than 2.
The digit in the hundreds place is odd.
Can you find the 2 numbers? Answered by Penny Nom. 





Forming 10 from 4 fours 
20081106 

From Corderro: I need a answer for 10 using only the number 4 Answered by Penny Nom. 





4 digit combinations 
20081105 

From Laura: I need to know all the number combinations possible with a 4 digit combination with double zero in the middle??? Answered by Penny Nom. 





Cutting a pipe on a 45 degree angle 
20080912 

From Bakshani: how do you a mark a pipe 5 inch diameter and cut it to form a 45 degree angel Answered by Stephen La Rocque. 





The game of 24 
20080820 

From Tasha: I have four numbers a 6, a 6 a 5 and a 3, i can add subtract divide or multiply these numbers in anyway but i have to use all of them and call only use them once i have to come up with a total of 24 Answered by Penny Nom. 





Four fours 
20080625 

From jen: Use four 4's and some of the symbols: +,,X, / and (), to give expressions for the whole numbers from 0 through9. Example : (4X4+4)/4=5 Answered by Janice Cotcher. 





Four digit combinations 
20080623 

From star: Can you you give me a list of all 4 digit combination starting with the number 7, for example 7123,7213. Answered by Penny Nom. 





A 4 sided lot 
20080603 

From Pat: If a lot of property is: front 127.87, back 156.17, one side
is
129.73, the other side is 125.00, then how much total acreage is
this. Answered by Harley Weston. 





WHY 144 where does this figure come from 
20080419 

From Sean: Why do you use the number 144 to divide
EG. length of timber 2 inches x 8 inches x 14 foot = 1.5 cubic feet
Calculation is 2x8x14 =224 divide by 144 = 1.5 WHY 144 where does this figure come from Answered by Leeanne Boehm. 





Some 4 sided polygons 
20080411 

From kathy: I am a 4th grade student and need to complete this question:
I need to draw 2 polygons with 4 sides and 2 right angles. Please help. Thanks Answered by Penny Nom. 





The game of 24 
20080410 

From sherwin: ok is like this you must get 24 using 1,5,5,5 but you can only use +,  , / and * Answered by Claude Tardif. 





The game of 24 
20080401 

From Frank: Hi, my name is Frank. I have been really stumped on this 24card. It has the 4 # of 21,3,5,10. If you have any suggestions, I would be grateful! Answered by Claude Tardif. 





A 4 digit passcode 
20080216 

From Rob: If there is a 4 digit passcode with only 6 punch in symbols, how many possibilities are there and what are they?
symbols are: Y, X, RB, RT, LB, LT. Please get back to me, thanks for your time and help. Answered by Stephen La Rocque. 





I am a 4digit number 
20080212 

From Nickie: I am a 4digit number with no repeating digits. I am divisible by 5, my first two digits (left to right) make a number divisible by 3, and my first three digits make a number divisible by 4. Also, my digits have a sum of 19 and I have the digit 7 in the thousands place. Who am I? Answered by Penny Nom. 





The game of 24 
20080206 

From Jameeka: What is a solution for 10,11,9 and 2 (Using fractions) in the game 24 square. Answered by Penny Nom. 





A 24 hour clock 
20080131 

From CARLA: at a certain time i looked at a 24hour digital clock and noticed some things about the time. (1 The number in the hours section plus the number in the minutes section makes 60.(2 One of the digits in the minutes section is the square root of the other digit in the minutes section.(3 The sum of the digits in the minutes section is the number in the hours section reversed. what is the time? Answered by Penny Nom. 





Choosing 10 numbers from 40 numbers 
20080118 

From Antwan: I have 40 numbers........number 140.
I want to know how many times i can chose 10 of those numbers without picking the same exact sequence twice if its even possible? Answered by Stephen La Rocque. 





The game of 24 
20080117 

From Raina: I have four numbers 3 3 9 5 that need to equal 24 using addition subtraction multipliction and division and I'm so stuck. Answered by Chris Langdon, Stephen La Rocque and Claude Tardif. 





Combinations 
20080108 

From david: is there a simple formula to follow for the number of combinations a set of numbers could have
ie.( lotto 649, 6 numbers, 149 range) Answered by Penny Nom. 





The game of 24 
20071119 

From M: Please help...how would you compute 2,2, 6, and 9 to equal 24? Answered by Penny Nom. 





The game of 24 
20071117 

From Todd: Hello. My son is very good at the Math 24 game and when he is stumped,
I am usually able to figure it out. However, we are both stumped with
this one. The numbers are 6, 6, 9 and 5. You can use addition, subtraction,
division and multiplication. Please help us equal 24 with those 4 numbers. Answered by Claude Tardif. 





Some four digit combinations 
20071025 

From Larisa: Please tell me the answer or how to get the answer of every possible digit combination of 5173, for instance1735, 1753, 1573, 1537, 1357, 1375, etc. Answered by Stephen La Rocque. 





The game of 24 
20070927 

From Amanda: I'm really stumped on a 24 math game question from math class. The numbers are 7,11,17,and 20. Help?? Answered by Claude Tardif. 





Four digit odd numbers 
20070926 

From sonia: how many 4 digit odd numbers can be made using the digits 2, 5, 7, and 8 Answered by Penny Nom. 





The largest 4digit odd number 
20070921 

From Courtney: what is the largest 4digit odd number you can make with 1  9. You can not repeat digits within a number Answered by Harley Weston. 





Four digit combinations 
20070904 

From Natalie: Using The Numbers 7 4 5
How many different 4digit combinations are possible, you can use any of the numbers twice. Answered by Penny Nom. 





The digital lock for a cell phone 
20070829 

From Hollie: i have a four digit lock code on my cell phone and cannot remember what it is. i know that it does not contain a 2 or a 4 and none of the numbers are repeated. the numbers range from 0 to nine and the only way to figure out what the code is, is to try randomly. I was wondering if you could give me a list of possible combinations using the information i gave you to make it easier to crack the code. Answered by Penny Nom. 





Four fours 
20070823 

From Belinda: My son came home with a math question that I can figure out. He needs to use only the number 4 and only four 4's, no less no more.
The problems need the answers 11,13,14,and 18. He can use any mathematical sign except square root. He's been able to get all but these
so far and I've worked them, but can't come up with the problems. Can anyone help? Answered by Stephen la Rocque. 





4 digit number combinations that start with the digit 2 
20070805 

From tonnie: my question is what are all the 4 digit number combinations that start with the number 2? Answered by Harley Weston. 





Four digit numbers 
20070730 

From Spence: My son has a problem that he has been unable to figure out. I was asked to
try, but I am confused on how to get the last two parts. Any help would
be great.
Thanks
Spence
How many fourdigit numbers can be formed under each condition?
A) The leading digit cannot be zero: 9(10)(10)(10)=9000
B) The leading digit cannot be zero and no repetition of digits
allowed: 9(9)(8)(7)=4536
C) The leading difit cannot be zero and the number must be less
than 5000: ??
D) The leading digit cannot be zero and the number must be even:
?? Answered by Penny Nom. 





24 Game: 7, 5, 3, 3 
20070611 

From Natasha: In the maths game called '24', how can you solve 7, 5, 3, 3?
I like playing a maths game called '24'. Answered by Stephen La Rocque. 





Game of 24 
20070404 

From teri: How do you find the answer of 24 by using 12,24,7 and 2 only once. Answered by Stephen La Rocque. 





The game of 24 
20070322 

From Britney: I have a math problem with the game 24. I am not sure how to solve it.
You can use +, , multiplication, division, ( ), square root, and exponents.
Please help me. The numbers are 1, 6, 7 and 8. You have to use all the
numbers and you can only use them once. Answered by Penny Nom. 





The game of 24 
20070309 

From colleen: We are playing the 24 math game where four numbers need to combine using any sign (+,  X, \) to equal 24. The four numbers are 22, 21, 6, 20. Answered by Penny Nom. 





Game 24 
20070209 

From Becka: I need help figuring out this sequence for 24 2 2 3 5 Please help Answered by Steve La Rocque, Penny Nom, Claude Tardif. 





14 golfers6 days 
20070202 

From tom: 14 of us are going golfing over the course of 6 days is there a formula that would allow us to have the chance to play with as many different partners as possible2 foursome, and two threesomes Answered by Chris Fisher. 





Four 9s 
20061215 

From A student: Can you make 67 with four 9's without using the ceilling method? Answered by Stephen La Rocque. 





The game of 24 
20061128 

From John: Math 24 with 9, 8, 4, 2? Answered by Stephen La Rocque and Penny Nom. 





How many digits are in the number 2^64? 
20060614 

From Trang: How many digits are in the number 2^{64}? Answered by Stephen La Rocque. 





More on the game of 24 
20060505 

From Kathy's daughter: My daughter's 5th grade class plays the 24 math game. She came home
the other day with a real stumper! The numbers on the card are 5,5,2,and 6.
The rules say you must use every number,you may only use a number once,you
can only add, subtract, multiply and/or divide and the answer is 24. Can you
help!!!! No one where I work has been able to figure this one. Answered by Paul Betts and Penny Nom. 





9, 8, 7 and 2 in the game of 24 
20060317 

From Alana: I just began student teaching in a 5th grade math class and found that the students play 24 in class, a game that I used to play in elementary school. The students choose a card which shows 4 numbers and then have to use any combination of addition, subtraction, multiplication, and division to reach 24. There was one card that we found that seems to have all of us stumped. The four numbers are 9, 8, 7 and 2. Can you figure out how to make 24 with those? Thanks so much for your help! Answered by Claude Tardif. 





More on the game of 24 
20060202 

From Cathy: 10,16,9,4 this numbers have to equal up to 24. we can't figure it out. Answered by Claude Tardif. 





The game of 24 with 7, 5, 5 and 4 
20051218 

From Nicole: i was playing the 24 game the other day. you get 4 digits and you have to turn them into 24. You can use addition, subtraction, division, and multiplication. You have to use all the numbers but you can only use each number once. an example is 2 4 4 1 the answer is (2x4)(41)=24. I got stuck on one. The numbers were 7 5 5 4 . Answered by Penny Nom. 





The game of 24 
20051010 

From Kim and Chris: My 4th grade son came home with a math game called 24 game. In it each card has 4 numbers on it. Each number can only be used once, in any order, using multiplication, subtraction, division, or addition. We are stuck on one and wondered if you could help. The four numbers are: 9, 9, 9, 9. And they must equal 24. Any clues? Answered by Claude Tardif. 





The game of 24 
20050110 

From Manny: HI  my daughter and I are being driven crazy by the following "24" question 7,7,3,3. She's good at the game (well, better than I am) and her teacher gave her this one. Any insight would be appreciated! Answered by Claude Tardif. 





The game of 24 
20040916 

From Angelica: how can you make 24 out of 7,7,6,5? Answered by Claude Tardif. 





The game of 24 
20040914 

From Alan: I was playing that math 24 game and i cant get 21,14,2,2 to equal 24 Answered by Claude Tardif. 





The 24 game 
20040909 

From Erik: Ive spent around 40 hours on this one problem. I need help. 3,3,7,7 Answered by Claide Tardif. 





The game of 24 
20040604 

From Samantha: i have the game 24 (the one where you have to use multiplication, division, addition, and subtraction to get to the answer 24) I was playing when i came across the 4 numbers 20, 17, 7, 11 and i cant seem to get 24. Answered by Claude Tardif. 





37 with four 4'2 
20040103 

From Myra: Using four fours and only four fours to get an answer of 37. Answered by Claude Tardif. 





Four digit combinations 
20031231 

From A student: i need all 4 digit combinations using 09 please Answered by Penny Nom. 





The 4th difference 
20031117 

From Jack: If i had a sequence which was the same at the 4th difference, aka difference
to the 4th, like:
1 5 14 30 56 96
(sequence)
4 9 16 26 40
5 7 10 14
2 3 4
1 1
then what would be the set formulars to find out a, b, c, d, and e
as i know the equation has to be an^{4}+bn^{3}+cn^{2}+dn+e Answered by Penny Nom. 





The game of 24 
20030903 

From A student: We have a game where four numbers must equal 24 using addition, subtraction, multiplication and/or division.Ý I am stuck on the following four numbers  12, 24, 9 and 17. Answered by Penny Nom. 





Two sequences that agree in the first 4 terms 
20030526 

From A student: Is it possible to have two formulas that define sequences that agree on the first four terms but not the rest? Answered by Penny Nom. 





Four digit palindromes 
20030505 

From Heather: How many palindromes were in between the numbers 1000 and 9999? Answered by Penny Nom. 





4 x abcd = dcba 
20021120 

From A student: Say you have a four digit number (e.g. abcd) and you multiply the number by 4. The answer you get will be the reverse order of the number you started with (dcba). What is the number? The four digits (a,b,c,d) cannot be the same number or cannot be repeated. Answered by Penny Nom and Claude Tardif. 





The 24 game 
20021116 

From Michael: Hello, my name is Michael and I am in elementary school .. We have a game we play in math class called 24 game. In it each card has 4 numbers on it. Each number can only be used once, in any order, using multiplication, subtraction, division, or addition. We are stuck on one and wondered if you could help. The four numbers are 109164. And they must equal 24. Any clues? Answered by Penny Nom. 





The repetition of doing many straight forward problems 
20021016 

From Dan: I am the father of a very bright 9 year old girl. She is very interested in math and she grasps concepts very quickly. I am concerned that she may become bored with the repetition of doing many straight forward problems. Her current math teacher has her working in a challenging math book and she is doing 3+ digit multiplication. She misses 2 or 3 out of ten of these problems due to simple addition errors. Should she continue to practice these problems until she can get them all correct? Or, should she move on? Answered by Kathy Nolan and Claude Tardif. 





Use a 4 four times 
20020912 

From Amanda: The question is what is if you can only use a 4 four times and get the answer 12, 10, and 6 Answered by Claude tardif. 





A schedule for 24 golfers 
20020525 

From John: I am working on a schedule for 24 golfers. 6 groups of 4. I have 8 golf days (twice per week for a month). Ideally, I would like to schedule all 24 golfers in 6 different groups of 4 on each day. Here is the catch.....no golfer in any group can be grouped togther more than once. Every group of 4 each day will have 4 new golfers that have never played together before. Is this possible? Answered by Chris Fisher. 





All four eights in three consecutive hands 
20020401 

From Blair: What are the odds against the same person holding all four eights in three consecutive hands. Answered by Brian Alspach. 





Some 5 card hands 
20020328 

From A student: From a standard deck of cards how many 5 card hands are possible consisting of a. exactly 4 hearts
b. two cards of one kind and three of another(like a full house). Answered by Penny Nom. 





100 from four 9's 
20020327 

From A student: My teacher gave us a math problem to try and figure out and I am stumped. The problem was this.... by only using four nines, how can you get an answer of 100? You can add, subtract, multiply or divide...anything you need to do ,but you can only use four nines. Answered by Leeanne Boehm. 





A lottery question 
20020325 

From Noel: I want to take all numbers in a Lotto game from 1 to 45 and ditribute them between 9 boxes each containing 5 numbers. Each number is used only once.  What is the probability that I derive the correct combination of 5 numbers in the draw?
 What is the probability if I take a systems game and now distribute 9 numbers into each of 5 boxes? Once again, the numbers are used only once.
 Can you explain what formulae are used and how the answer is derived? Is this a distribution or combination problem? Or is it a combination of both distribution and combination?
Answered by Andrei Volodin and Penny Nom. 





Double square roots 
20020217 

From Ali: i have a question about how to do double square roots with variables and powers. example : v/"" v/"" 81y^{8} Answered by Harley Weston. 





Making 24 
20020117 

From Renee: My 4th grade daughter and I need to find a simple math sentence using 5, 5, 3, & 7 to equal 24. You can add, subtract, divide or multiple. Answered by Penny Nom. 





450 students 
20011219 

From Jessie: My teacher told me that the number of ways of lining up the 450 students in our school is larger than the number of atoms in the universe. Is this right? This question came to Good Question, a regular feature on the CBC Morning Edition radio show in Saskatchewan. Answered by Denis Hanson and Harley Weston. 





4 digit combinations 
20010922 

From Laura: I need all of the 4 digit combinations using the numbers 09. Answered by Harley Weston. 





Five fourths 
20010408 

From Linda: Can you have 5/4 of something? Answered by Penny Nom. 





4160 
20010307 

From Jessica: The product of two consecutive page numbers in this book is 4160. What are the page numbers? Answered by Penny Nom. 





Lotto 649 
20010227 

From A student: I am a secondary student and I was wondering if there was a better way to find out how many combinations and what they are in the lotto 6/49 than writing them out on a piece of paper. Any help you can provide me with would be greatly appreciated. Answered by Penny Mon. 





Math game 24 
20010221 

From Kristi: We have a game we play in math class called 24 game. In it each card has 4 numbers on it. Each number can only be used once, in any order, using multiplication, subtraction, division, or addition. We are stuck on one and wondered if you could help. The four numbers are: 3, 3, 5, 7. And they must equal 24. Any clues? Answered by Claude Tardif. 





A four dimensional object 
20010214 

From A student: Can you give me some examples of a four dimensional object you can find around your house? Answered by Harley Weston. 





Lucky dip 
20010209 

From Andrew: I buy 8 lucky dip (random number) panels, each panel has 6 numbers chosen at random from 49. I never seem to cover more than 32 different numbers on average, over the total 48 numbers chosen, which seems low? How can I calculate the probability of 32 different numbers or 33 diferent numbers? Answered by Claude Tardif. 





Divisibility by 16 
20001212 

From Shiling: A number can be divided by 16 if and only if its 1st four digits can be divided by 16. How can you prove that? Answered by Penny Nom. 





1234567890 
20000914 

From Bradley Kloetzly: Can you find two whole numbers, with the smallest possible difference between them, which when multipled together equal: 1234567890? Answered by Harley Weston. 





649 
20000914 

From Steve: In our state lottery you must choose 6 numbers (149). How many different combinations are there? They can be in any order. Answered by Harley Weston. 





4digit numbers using 7, 8, 9 and 0 
20000913 

From Ryan: Could you please help me with the following problem, How many different 4digit numbers are there that use each of the digits 7,8,9,and 0? Please list them. Answered by Penny Nom. 





4 Color map Theorem 
20000602 

From Rick Slen: I am challenging my students to find a map that can not be coloured using only 4 colours. I know that it is not possible, but one boy drew a map with a "country" totally surrounding all the others so of course it touched all other colours. How do I explain that this type of map is not permissable? Answered by Denis Hanson. 





Is a sheet of paper 2dimensional? 
20000306 

From Jaclyn Evans: One day, we were discussing 3dimensional items. Is a sheet of paper considered a 3dimensional shape? Or is it 2dimensional as described by 8.5 by 11 size? Also, could you give me an example of a 4dimensional item or scenario? Answered by Harley Weston. 





A 4by4 magic square 
20000206 

From Maureen Fitzsimons: I need to create a 4x4 grid using numbers .1, .2, .3, .4, ....1.1, 1.2, 1.3,1.4,1.5,1.6 the sum of the number diagonally, horizontally and across all equal 3.4 Answered by Penny Nom. 





.400 in Baseball 
19991215 

From Nieve Nielson: I have several questions to ask about the probability of getting a .400 in baseball:  What is the probability of a baseball player hitting a .400 in one game, considering that the hitter is up to bat four times?
 What is the probability of a baseball player hitting a .400 in a season, considering that there are 100 games in a season? With the first two questions in mind:
 Considering that baseball has been around for about 100 years, about how many people should hit .400 inthat time?
Answered by Chris Fisher. 





2,5,8,1 yields 24 
19990912 

From Terry Hill: I was wondering how do I make the numbers 2,5,8,and 1 equal 24? Answered by Penny Nom. 





Measuring the fourth dimension 
19990607 

From Suzanne Hall: Hi! The other day, two of my students asked a very interesting question: Is the fourth dimension measured with hypercubes? Their reasoning went like this: Lines are 1D and are measured with line segments, which are part of a line. Planes are 2D and are measured with squares, which are part of a plane. Space is 3D and is measured with cubes, which are part of space. So, logically, hypercubes would be used to measure the fourth dimension. Answered by Chris Fisher. 





Factoring 
19990308 

From L. Sivad: Question: m^{2}+6m+9n^{2} Answered by Penny Nom. 





Grade 4 curriculum 
19990209 

From John: My daughter is in grade 4 and her math teacher has asked her to write out the number 1 to 10,000. She is currently at 4567 and is sick of it. When I complained that this was not a usefull exercise she said that the Ministry of Education curriculum requires this and gave a photocopy of the page. It states: "read and write whole number to 10,000 in standard, expanded, and written forms (eg., 9367=9000+300+60+7 = nine thousand three hundred sixtyseven). Has the teacher interpreted this the wrong way? Thanks John PS. Do you have a web address for Ontario Math and Science Teachers organization? Answered by Jack LeSage. 





3 to the power of 1994 
19980905 

From Kim Tangney: What are the last two digits of:  3 to the power of 1994
 7 to the power of 1994
 3 to the power of 1994 + 7 to the power of 1994
 7 to the power of 1994  3 to the power of 1994
Answered by Penny Nom. 





x^2 = ...444 
19980223 

From James Bauer: What is the first interger that when squared ends in three 4's? (ex. x^2 = ...444) Prove that there are no intergers that when squared end in four 4's (ex. x^2 = ...4444) Answered by Penny Nom. 





1 exposant 4 
20041219 

From Denis: ma question est : 1 exposant 4 = ? Quand nous avons une base négative sans parenthèse comme ma question Answered by Diane Hanson. 

