# 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 late June.

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

##### Problem #062 – sliding coins

Can you align all of the coins on the right edge of the board?

##### Problem #061 – flower garden

Three mathematicians discuss a beautiful flower garden and the coloured flowers within.

##### Problem #060 – realtor commissions

Two realtors discuss who's netting the award for highest average commission, but it isn't clear who the winner is...

##### Problem #059 – marching pegs

How can you swap the coloured pegs if they can only march forward?

##### Problem #057 – how to find the circle centre

Can you find the centre of the circle with just five lines?

##### Problem #056 – tennis tournament

How many matches does it take to find the winner of a tennis tournament?

##### Problem #055 – horse racing

25 horses racing, and you have to find out the fastest ones!

##### Problem #054 – imperfect compression

Can you show that perfect compression is impossible?

##### Problem #053 – weighing the odd one out

Can you find the fake ball by weighing it?

##### Problem #052 – chessboard domino

Can you tile a chessboard with two missing squares?

##### Problem #051 – queens and knights

How many queens and knights can you place on a chessboard?

##### Problem #050 – 8 queens

In how many ways can you place 8 queens on a chessboard?

##### Problem #049 – coin pyramid

Can you make the pyramid point the other way by moving only three coins?

##### Problem #048 – trick or treat

Can you help these kids trick or treat their entire neighbourhood in this Halloween special?

##### Problem #047 – surgery gloves

How can two doctors operate two patients with only two pairs of latex gloves?!

##### Problem #046 – triangle grid

Can you draw 4 triangles in this 5 by 5 grid, covering all dots?

##### Problem #045 – 1, 2, 3

Figure out the number I'm thinking of with a single question!

##### Problem #044 – send more money

Can you solve this simple-looking arithmetic challenge?

##### Problem #043 – Rubik's cube scrambling

If I scramble a Rubik's cube for long enough, will it solve itself?

##### Problem #042 – mine captcha

Can you solve this little minesweeper puzzle?

##### Problem #041 – canyon crossing

It's night time and 4 friends need to cross a fragile bridge, but they only have one torch. What's the order in which they should cross?

##### Problem #040 – the dozen puzzle

Three friends are given three different numbers that add up to a dozen. Can you figure out everyone's numbers?

##### Problem #039 – rope timer

You have two magical ropes that you can set on fire and you need to count 45 minutes. How do you do it?

##### 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?

##### Problem #037 - game of coins

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

##### Problem #036 - huge, tiny triangle

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

##### Problem #035 – unknown coin sides

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

##### 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?

##### 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?

##### 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?

##### 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?

##### 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?

##### Problem #029 - hidden key 2 🗝️🗝️

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

##### 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?

##### 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..?

##### 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!

##### 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?

##### 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?

##### Problem #023 - guess the polynomial

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

##### 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.

##### 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!

##### 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.

##### 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!

##### Problem #018 - circle of hats

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

##### 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?

##### 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.

##### 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!

##### 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!

##### 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..?

##### Problem #002 - a bag full of numbers

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

##### 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.