### Brain Teasers

# One Hundred People at a Party

Exactly one hundred people were at a party. Each either always tells the truth or always lies. After the party, one of the one hundred people said, "I shook hands with exactly one truthteller at the party." Another person says, "I shook hands with two truthtellers," and a third says, "I shook hands with three." And so on until someone says, "I shook hands with 99 truthtellers." The hundredth and final person says, "I shook hands with no truthtellers at all!" How many truthtellers were at the party?

### Answer

There was only one truthteller.Let the person who said he shook hands with one truthteller be Mr. One, the one who said he shook hands with two truthtellers be Mr. Two, and so on.

Suppose Mr. Ninety-Nine is a truthteller. This means the others are all truthtellers and he shook hands with them all. But if Mr. Zero and Mr. Ninety-Nine are both truthtellers, Mr. Zero would not have shaken hands with Mr. Ninety-Nine. So Mr. Ninety-Nine must be a liar.

Suppose Mr. Ninety-Eight is a truthteller. This means he shook hands with everyone else except Mr. Ninety-Nine. But if Mr. Zero and Mr. Ninety-Eight are both truthtellers, then Mr. Zero would not have shaken hands with Mr. Ninety-Eight. So Mr. Ninety-Eight must also be a liar.

Using similar logic, we can determine that everyone between Mr. Ninety-Seven and Mr. One is a liar.

What about Mr. Zero? Since everyone else was a liar, he really did not shake hands with any truthtellers. Therefore, Mr. Zero is the only truthteller.

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## Comments

Terrific. Perfect little logic teaser. A fun example of iterative or recursive thinking.

Keep 'em comin' WimpyKid! I love these!

Keep 'em comin' WimpyKid! I love these!

I never would have thought of that

Just to let you guys know a new teaser of mine (for Simpsons fans) was just accepted.

I am a truth teller. This teaser was confusing to me so I did not care for it. I hope your next one is better, but glad the others liked it.

I'm with Babe. I do not like these types of teasers.

Are we both telling the truth, are we both lying? Who even knows...

Are we both telling the truth, are we both lying? Who even knows...

Not my cup of tea.

Mr. One shook hands with 1 truth teller, Mr. Zero, and said he shook hands with 1 truth teller. therefore he told the truth and is a truth teller.

This puzzle has no answer.

This puzzle has no answer.

Wow. Mixed feelings here. I love this puzzle!

I am very excited to see that it was chosen as a Teaser of the Day. Kudos to the ongoing support of the site volunteers bringing these new puzzles into the Daily Teaser mix. As shown in the comments, the Daily Teaser is the primary way puzzles get exposed.

I'm disappointed this was not received well by the commenting audience at large. I'm not disparaging anyone. We all have our favorite types. The math/logic people seem to be in hiding.

Mr. One said he shook hands with one truthteller. He lied. He may not have shaken hands at all, or he may have shaken hands with any of 98 other partygoers. The logic in the teaser is sound.

Three cheers for a good logic Daily Teaser!!

I am very excited to see that it was chosen as a Teaser of the Day. Kudos to the ongoing support of the site volunteers bringing these new puzzles into the Daily Teaser mix. As shown in the comments, the Daily Teaser is the primary way puzzles get exposed.

I'm disappointed this was not received well by the commenting audience at large. I'm not disparaging anyone. We all have our favorite types. The math/logic people seem to be in hiding.

Mr. One said he shook hands with one truthteller. He lied. He may not have shaken hands at all, or he may have shaken hands with any of 98 other partygoers. The logic in the teaser is sound.

Three cheers for a good logic Daily Teaser!!

I agree that the logic is sound, and I enjoyed the teaser. Thanks for posting it.

Bondline: As long as Mr One did not shake hands with Mr Zero, then Mr Zero is the only truth teller, and all the others will be liars no matter how many other peoples hands they shook. The puzzle did not say that everyone shook hands with everyone. Its possible no one shook hands at all (in that case its obvious who is telling the truth and who is a liar)

This only works if you require that everyone at the party shakes hands with everyone else. Otherwise they could all be truth-tellers since the number of truth-tellers they shook hands with would be precisely the number of people they shook hands with (the last one telling the truth because he didn't shake hands at all). Without the stipulation that everyone shakes hands with everyone else, in fact, all values from 1 to 100 are potentially valid answers. The only value that does not work is zero since the last person would have to be telling the truth if the other 99 people were liars. The puzzle as stated is indeterminate.

Just to let you know I have two more teasers coming right up!

Beautifull teaser!! Solving it went rather easy, but to come up with this is magnificent!

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