Did you solve it? The poker puzzle that has everyone fooled

Earlier today I set you these two puzzles, which are extracts from my new book Think Twice: Solve the Simple Puzzles (Almost) Everyone Gets Wrong. Here they are again with solutions.

1) Pint-sized problem

Which is longer? a) the height of this pint glass, or b) the circumference of its rim?

Comment: you may have seen this question in the context of classic UK pint glasses, but I am asking about continental style pint glasses, which are much thinner and higher.

Solution b) The rim is longer.

Next time you are in a pub, ask anyone this question. They will invariably say the height. But with a traditional pint glass, the circumference is almost twice as long as the height. Even with this extremely svelte Peroni glass, the rim nudges it. It doesn’t look like it is true but it is.

The circumference of a circle is pi (3.14 to two decimal places) times the diameter. The diameter of a Peroni glass is 76mm, giving a circumference of 238mm. The height is a mere 231mm.

The moral of the story: humans are very bad at intuiting circumferences. We are used to measuring lengths as straight lines. When we see a glass side on, we look left to right and back again. But the circumference is not double the diameter – it is more than three times as long. Dont forget the pi in pint!

2) Fool House

You are playing poker. Which is the stronger of the two hands below?

For clarification, the highest ranked poker hands are, in order from the strongest, royal flush , straight flush, four-of-a-kind, and full house . When two hands of the same rank face off, the winner is the hand with the cards of highest value. (For a more detailed description of the ranks see below.)

Solution b) is stronger.

People here fall into the trap of thinking that I am asking which hand is stronger if the hands are playing against each other. If that were the case, the first hand would be stronger because it has the higher value cards.

However, they cannot be playing against each other! Poker uses a single deck and there are only four aces is total. If one player has the first hand, no one has the second hand, and vice versa.

The meaning of ‘stronger’ in the context of the puzzle is the hand that puts a player in the stronger position in a game. Perhaps a more concise setting of the puzzle is ‘Which hand would you rather be dealt?’

Both hands are equally strong against other full houses and four-of-a-kinds. But AAA99 blocks more straight flushes, since it means an opponent cannot have 5-9, 6-10, 7-J, 8-Q or 9-K. The AAAKK only blocks 9-K.

The effect is only marginal (perhaps a reader will want to calculate the percentages) but it is there!

I hope you enjoyed these puzzles. If you did, Think Twice has many more. The book is just out, and – dare I mention Christmas? – it would make a perfect gift. The puzzles are designed to be fun when solving as a group.

Thanks to Shoma Amin from Asahi UK.

Poker definitions: a royal flush is a run of AKQJ10 in a single suit, a straight flush is any other run of consecutive cards in the same suit, a four-of-a-kind is a hand with all four cards of the same value, and a full house is two of one value and three of the other.

Think Twice: Solve the simple puzzles (almost) everyone gets wrong (Square Peg, £12.99). To support the Guardian and Observer, order your copy at guardianbookshop.com . Delivery charges may apply.

I’ve been setting a puzzle here on alternate Mondays since 2015. I’m always on the look-out for great puzzles. If you would like to suggest one, email me .