Combinatorics intro problem 2 - solution

If you can still remember this post, we left you with another combinatorics problem. In reality, this second one was more of a brainteaser than an Olympiad style problem. So I'll use more pictures and fewer words here.

Here's the question again:

Two friends, Ann and Bill, are shown three pieces of paper, two pieces have the number 1 on it, and one piece has the number 2 on it. One piece is now attached to Ann's back and one to Bill's back; both Ann and Bill have the number 1 on their backs, and the piece with number 2 is hidden. Each person can see the other's number but not their own number, and neither Ann nor Bill know what we put on their back and which number we hid. Assuming that both the contestants are good at problem solving, can one of them be the first to deduce logically (not guess) what number is on his/her back?

And my customary picture:

My attempt to draw people made them look like lollipops.

My attempt to draw people made them look like lollipops.

Let's consider an imaginary situation. What if one of the people (let's say Ann, for sake of explanation) had a "2" on their back?

"Gasp!!" says the lollipop on the right.

"Gasp!!" says the lollipop on the right.

Bob will see the "2" on Ann's back, and realise that there is only one "2". That means that he can't have a "2" - he has a "1", so he immediately jumps up and down crying with joy for having worked out what that little piece of paper on his back said.

Now let's get back to reality. Neither Ann or Bob see a "2" on the other person's back, so they stay a bit quiet for a bit, twiddling their thumbs.

It was hard enough drawing arms and legs. I'm not going to draw thumbs.

It was hard enough drawing arms and legs. I'm not going to draw thumbs.

But what if one person realises this: if he/she had a "2" on his/her back, the other person would have realised their own number and jumped around happily! So the fact that the other person isn't celebrating means... that he/she (the person who realised this) doesn't have a "two" on his/her back! Bend those minds.

Here are a few things to think about:

  • Which person will realise this first? The smarter person?
  • Let's say Ann celebrates when she discovers their own number. From Bob's perspective, is this because Ann saw a "2" on Bob's back, or because she deduced that she must have a "1" based on all this logic? Is there any way for Bob to work out his own number then? What if Bob discovered his number first?
  • What if Ann thinks that Bob is slow, so when he sees a "2" on her back, he doesn't realise that it means he has a "1" on his own back?
  • I just realised that the problem didn't restrict Ann and Bob from communicating in any way. But if both of them are trying to race each other to find their own number first, the problem becomes interesting again...

Too many words. I'll just stop here.

BC