MHB Combinatorics - The pigeonhole principle

[Yankel]

Hello,

I am trying to solve a problem related to natural numbers. The solution is based on the pigeonhole principle, however I can't see the connection.

The is the problem:

Choose 12 two digit numbers. Divide each by 11 and write down the residue (i.e. do the modulu operation). Group the residues in different sets, in such a way that all numbers with the same residue are in the same set.
Can you find two numbers that when subtracted from one another (bigger - smaller) gives a two digit number with identical digits ? (e.g. 57-24=33).

Now choose a new set of 12 numbers. Can you find such numbers now ?

Are you findings random or is there a reason ? Try to write down a rule and to prove it.

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So I have chosen 12 numbers and did all the residues. I have noticed that two numbers in the same set, i.e. two numbers having the same residue will give a subtraction which is a number with identical digits.

What I don't see, is what's the connection to the pigeonhole principle, what is the rule I am suppose to find and how to prove it using the pigeonhole principle.

Thank you in advance !

P.S.

My chosen example was:

{33} {12} {24, 57} {25} {81} {17,39} {73,95} {41} {64}

 

[Yankel]

I think I figured something here, not the whole story yet.

Two numbers will give the same residue will be in the same set. Numbers from the same set will give a two digit number with the same digit when subtracted.

If I divide by 11, then I have 11 different residues (0-10). When picking 12 numbers, I guarantee that at least one set will have two numbers in it, and then according to the pigeonhole principle, I will have two numbers giving the required result when subtracted.

The only thing missing here, is why two numbers in the same set give a number with identical digits when subtracted ? Is there a way to know which number it will be (e.g., 22, 33, ...) ?

Can I say that I will ALWAYS get two numbers that will give number with identical digits when subtracted ? No matter which 12 numbers I choose ?

A new discovery...all differences I got are divided by 11 with no residue...

 

[Evgeny.Makarov]

A number between 1 and 99 (the difference of two numbers with equal remainders) is a multiple of 11 iff it has two identical digits.