
Date: Mon, 20 Sep 1999 07:22:02 0600 (CST)
Name: Jessica
Who is asking: Student
Level: Elementary
Question:
Is there a trick to finding out how many numbers containing the digit two is there from 1 to 1000? Please Explain.
Hi Jessica,
 Lets start with the largest group:
All the two hundreds: that gives 100 numbers containing two.
 Now, in every OTHER group of 100, there are 10 other
numbers for the twenties:
10 x 9 (nine other hundreds groups).
 Finally, in every group of tens which you did NOT yet count,
there is a number with last digit 2.
I presume you can figure
that out (by addition and substraction).
Finally you add the counts you made in 1, 2 and 3 since
you already took care of any overlap.
In a sense, as I write this I am imagining a long list of all the
numbers (in my mind) and putting down marks by all the numbers
with 2's in the hundreds, then the tens and finally the units.
With this picture, I see that one set of 20 lies in the 200's,
so I don't count these twice. Other sets of 2's lie within
20's of each hundred, so I am carefull not to count these twice.
 1  2  3  4  5 
6  7  8  9 
10  11  12  13  14  15 
16  17  18  19 
20  21  22  23  24  25 
26  27  28  29 
30  31  32  33  34  35 
36  37  38  39 
.  .  .  .  .  . 
.  .  .  . 
.  .  .  .  .  . 
.  .  .  . 
110  111  112  113  114  115 
116  117  118  119 
120  121  122  123  124  125 
126  127  128  129 
130  131  132  133  134  135 
136  137  138  139 
.  .  .  .  .  . 
.  .  .  . 
.  .  .  .  .  . 
.  .  .  . 
200  201  202  203  204  205 
206  207  208  209 
200  211  212  213  214  215 
216  217  218  219 
220  221  222  223  224  225 
226  227  228  229 
.  .  .  .  .  . 
.  .  .  . 
.  .  .  .  .  . 
.  .  .  . 
290  291  292  293  294  295 
296  297  298  299 
300  301  302  303  304  305 
306  307  308  309 
.  .  .  .  .  . 
.  .  .  . 
.  .  .  .  .  . 
.  .  .  . 
There are fancy 'counting methods' which one can use
(and loearn) in higher mathematics, checking out union and
intersection with addition and substraction. However, our
visual sorting works pretty well for patterns like you describe.
Walter Whiteley
Go to Math Central

