Given 12 different 2-digit numbers, show that one can choose two of them such that their difference is a two-digit number with identical first and second digit

Answers

Step-by-step explanation:

there are 91 2-digit numbers : 10 ..99

and there are 9 2-digit numbers with identical first and second digit : 11, 22, 33, 44, 55, 66, 77, 88, 99

99 has to be excluded, because there are no 2 2- digit numbers that I can subtract from each other and get 99.

so, e are dealing with 8 possibilities.

88 = 99-11

98-10

77 = 99-22

98-21

97-20

…

78-1

66 = 99-33

98-32

…

67-1

…

11 = 99-88

98-87

…

12-1

all the double- digit numbers are 11 apart. so, by picking 12 numbers, I have to have at least one that I can combine with one of the other 11 to get a double-digit number.

Step-by-step explanation: