Problems

Blog posts with problems to get your brain going! You get a new problem every fortnight and the solutions are published later.

If you want the problems delivered to your inbox be sure to subscribe to the Problems newsletter.

The next problem is scheduled for Sunday, 27th of June.

You may notice some problems are missing... I'm still migrating articles from my old blog.

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Problem #038 – bridges of Königsberg

You are on vacation and must find the most efficient way to cross all bridges. How will you do that?

111
Problem #037 - game of coins

Alice and Bob sit across each other, ready for their game of coins. Who will emerge victorious?

248
Problem #036 - huge, tiny triangle

Can you find a really large triangle that is also really tiny?

233
Problem #035 – unknown coin sides

This is an algorithmic puzzle where you just have to turn some coins.

559
Problem #034 – one question to happiness

Two doors, one gives you eternal happiness and the other eternal sadness. How can you pick the correct one?

403
Problem #033 - syncro

Syncro is a beautiful game where you have to unite all the petals in a single flower. In how many moves can you do it?

746
Problem #032 - restaurant roundtable

A waiter at a restaurant gets a group's order completely wrong. Can you turn the table to get two or more orders right?

670
Problem #031 - piped ants

A bunch of ants are left inside a very, very, tight tube, and they keep colliding with each other and turning around. How long will it take them to escape?

852
Problem #030 - efficiency at the beach

You are sunbathing when you decide to go and talk to some friends under a nearby sun umbrella, but first you want to get your feet wet in the water. What is the most efficient way to do this?

852
Problem #029 - hidden key 2 🗝️🗝️

This problem is a step up from Problem #028 - hidden key. Can you tackle this one?

902
Problem #028 - hidden key 🗝️

There is a key hidden in one of three boxes and each box has a coin on top of it. Can you use the coins to let your friend know where the key is hiding?

593
Problem #027 - pile of coconuts 🥥

Five sailors and their monkey were washed ashore on a desert island. They decide to go get coconuts that they pile up. During the night, each of the sailors, suspicious the others wouldn't behave fairly, went to the pile of coconuts take their fair share. How many coconuts were there in the beginning..?

1034
Problem #026 - counting squares

I bet you have seen one of those Facebook publications where you have a grid and you have to count the number of squares the grid contains, and then you jump to the comment section and virtually no one agrees on what the correct answer should be... Let's settle this once and for all!

696
Problem #025 - knight's tour

Alice and Bob sit down, face to face, with a chessboard in front of them. They are going to play a little game, but this game only has a single knight... Who will win?

920
Problem #024 - hats in a line

Some people are standing quiet in a line, each person with a hat that has one of two colours. How many people can guess their colour correctly?

656
Problem #023 - guess the polynomial

In this problem you have to devise a strategy to beat the computer in a "guess the polynomial" game.

585
Problem #022 - coprimes in the crowd

This simple problem is an example of a very interesting phenomenon: if you have a large enough "universe" to consider, even randomly picked parts exhibit structured properties.

897
Problem #021 - predicting coin tosses

Alice and Bob are going to be locked away separately and their faith depends on their guessing random coin tosses!

801
Problem #020 - make 24 with 3 3 8 8

Today we are visiting a specific instance of a well-known basic mathematics game, the 24 Game. The "24 Game" is usually played with younger students because it helps them develop skills related to the basic arithmetic operations.

1021
Problem #019 - fold the alphabet

Take out a piece of paper and a pencil, I am going to ask you to write some letters in your sheet of paper and then I am going to challenge you to fold the sheet of paper... with a twist!

975
Problem #018 - circle of hats

$$n$$ mathematicians with numbered party hats gather around in a circle... It is a matter of life or death!

951
Problem #007 - binary multiples

Is it true that every integer you can think of has a multiple written out only with $$0$$s and $$1$$s?

478
Problem #006 - stacks of beans

I find the problem in this post rather fun to think about because it is a problem about a game that can actually be played between two players.

688
Problem #005 - number me right

This post's problem is a really interesting problem I solved two times. The first time I solved it I failed to prove exactly how it works... then some years later I remembered the problem statement and was able to solve it properly. Let's see how you do!

728
Problem #004 - solvability of the water buckets

In this post I talked about the riddle of the water buckets. Now I challenge you to prove that in some situations it is impossible to solve it!

845
Problem #003 - a quarrel in the Shire

Gandalf has some Hobbits to appease but his task seems to go on forever. Can you give him a hand..?

741
Problem #002 - a bag full of numbers

Two friends were bored and decided to play a game... a mathematical game with a paper bag!

1030
Problem #001 - a dancing triangle

This post's format will be a bit different from the usual and the first of a series of posts of this type. In this post, I will state a problem and then present my solution.