# each of the students in a class writes a dirrerent 2 digit number on the whiteboard. the teacher claims that no matter what the students write, there will be at least three numbers on the whiteboard whose digits have the same sum. what is the smallest number of students in the class for the teacher to be correct?

I got the answer of 31 students. am i right? if i am not,could someone please show me how to figure this out?

Look at the sums:

ways to get sum of 1...10

ways to get sum of 2....20,11

How can three students get 3> 2,0;1,1, disallow 0,2

How can three students get 4> 4,0;22, 13, 31

get 5? 32, 23, 14, 41, 50

get 6 60, 51,15,24,42, 33

get 7 25, 16, 52, 61, 70, 43,34

get 8 17,71, 80, 62, 26, 35, 53,44

get 9 45, 54, 63,36, 27,72,`18,81,90

get 10 19,91,28,82,37,73,46,64, 55

get 11 29,92,38,83,74,47,56,65

notice the pattern...the ways to get a sum is always Sum until 9, then it is receeding in order.

get 12 ...7 ways (39,93,48,84,57,75, 66

get 13 six ways

get 14 five ways

get 15 four ways

get 16 three ways

get 17 two ways

get 18 one way.

Ok, then to have three in one sum, you will have two single sums, 15 of two sums, and one of three sums.

Number kids: 2 + 15*2 + 1*3

check my thinking.