# 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.

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1. 31 students

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2. In 18 ways get the sum from 1 to 18, I think if 3 students choose one any way of these sums, the teacher is correct. So the number of the students is 17x2+1x3=37

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