Difficulty – Medium

Medium difficulty brain teasers and puzzles

Same Number of Handshakes

In some cultures, it is typical for guests at an event to shake hands when they meet other guests (when there isn’t a pandemic). Some might be more social and shake hands with a lot of other guests, while others may be less social and shake hands with few or no other guests.

Can you prove that, regardless of the number of guests at the event, there must be at least two guests with the same number of handshakes at the event? In other words, can you prove that it is impossible for every guest to have shaken a different number of hands?

Assume guests can’t shake hands with themselves.

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Middle Four Letters

The middle four letters of these 8-letter words are shown. Can you figure out the full 8-letter words? There may be multiple suitable answers for some of the words.

  1. _ _ EQUA _ _
  2. _ _ DUST _ _
  3. _ _ CUBA _ _
  4. _ _ COLA _ _
  5. _ _ DIRE _ _
  6. _ _ MESA _ _
  7. _ _ TACO _ _
  8. _ _ OPIC _ _
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Inscribed Squares

Squares Inscribed in Right Triangles

Above are two identical isosceles right triangles containing two inscribed squares.

In one, a perfect square has been inscribed such that two sides line up with the two legs of the right triangle. In the other, a perfect square has been inscribed such that one side lines up with the hypotenuse of the right triangle.

Is the inscribed square on the left larger or the one on the right?

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Card Flipping Endgame

There are n playing cards lined up face-down in a row. Every turn, a pair of adjacent cards with the left card face-down is randomly selected (i.e., a pair of cards has no chance of being selected if the left card is face-up, otherwise all pairs are equally likely to be selected). Both cards are then flipped over (face-down to face-up or face-up to face-down).

Prove that after enough turns, it will eventually be impossible to select a pair of cards with the left card face-down.

When you reach this card flipping endgame, will the rightmost card be face-up or face-down?

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