Digit Spans

A puzzle by A. Savin: Using each of the digits 1, 2, 3, and 4 twice, write out an eight-digit number in which there is one digit between the 1s, two digits between the 2s, three digits between the 3s, and four digits between the 4s.

Click for Answer

The Jeweler’s Observation


Prove that every convex polyhedron has at least two faces with the same number of sides.

Click for Answer

Black and White

ebert chess problem

By Hilmar Ebert. White to mate in two moves.

Click for Answer

Ride Sharing


You and I have to travel from Startville to Endville, but we have only one bicycle between us. So we decide to leapfrog: We’ll leave Startville at the same time, you walking and I riding. I’ll ride for 1 mile, and then I’ll leave the bicycle at the side of the road and continue on foot. When you reach the bike you’ll ride it for 1 mile, passing me at some point, then leave the bike and continue walking. And so on — we’ll continue in this way until we’ve both reached the destination.

Will this save any time? You say yes: Each of us is riding for part of the distance, and riding is faster than walking, so using the bike must increase our average speed.

I say no: One or the other of us is always walking; ultimately every inch of the distance between Startville and Endville is traversed by someone on foot. So the total time is unchanged — leapfrogging with the bike is no better than walking the whole distance on foot.

Who’s right?

Click for Answer

A Passing Wave


A puzzle from J.A.H. Hunter’s Fun With Figures (1956):

A man paddling a canoe upstream sees a glove in the water as he passes under a bridge. Fifteen minutes later, he turns around and paddles downstream. He passes under the bridge and travels another mile before reaching the rock from which he started, which the glove is just passing. If he paddled at the same speed the whole time and lost no time in turning around, what is the speed of the current?

Click for Answer

Modern Art

modern art puzzle

Which part of this square has the greater area, the black part or the gray part?

Click for Answer

Black and White

shinkman chess problem

By William Anthony Shinkman. White to mate in two moves.

Click for Answer

Roll Call

A problem from the 2002 Moscow Mathematical Olympiad:

A group of recruits stand in a line facing their corporal. They are, unfortunately, rather poorly trained: At the command “Left turn!”, some of them turn left, some turn right, and some turn to face away from the corporal. Is it always possible for the corporal to insert himself in the line so that an equal number of recruits are facing him on his left and on his right?

Click for Answer

Podcast Episode 56: Lateral Thinking Puzzles


Here are six new lateral thinking puzzles to test your wits! Solve along with us as we explore some strange scenarios using only yes-or-no questions. Many were submitted by listeners, and most are based on real events.

A few associated links — these spoil the puzzles, so don’t click until you’ve listened to the episode:

Puzzle #1

Puzzle #3

Puzzle #4

You can listen using the player above, download this episode directly, or subscribe on iTunes or via the RSS feed at http://feedpress.me/futilitycloset.

Please consider becoming a patron of Futility Closet — on our Patreon page you can pledge any amount per episode, and all contributions are greatly appreciated. You can change or cancel your pledge at any time, and we’ve set up some rewards to help thank you for your support.

You can also make a one-time donation via the Donate button in the sidebar of the Futility Closet website.

Futility Closet listeners can get $5 off their first purchase at Harry’s — enter coupon code CLOSET at checkout.

If you have any questions or comments you can reach us at podcast@futilitycloset.com. And you can finally follow us on Facebook and Twitter. Thanks for listening!

Black and White

gold chess problem

By Sámuel Gold. White to mate in two moves.

Click for Answer