Date: Mon, 20 Sep 1999 07:22:02 -0600 (CST)
Name: Jessica
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,

All the two hundreds: that gives 100 numbers containing two.

2. Now, in every OTHER group of 100, there are 10 other numbers for the twenties:
10 x 9 (nine other hundreds groups).

3. 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

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