Foul winds are bringing tidings of ill will. You need to gather your clansmen. You have one trusted man servant who will go recruiting for you. If you send him out on a five day journey – five clansmen will start to arrive on the fifth day and every day thereafter.

You win if on any day your forces outnumber your opponent’s forces by… let’s say 30.

Unfortunately, your evil opponent discovers that you are sending out your trusted man servant for five days. How many days will she send out her trusty man servant?

She could choose 10 days.

On day nine you are about to win – you have 25 men… She still has none! Remember – the winner is the person who gets 30 or more men more than their opponent. Unfortunately – on day 10 She has 10 and you have 30 – a difference of only 20 ;-(

Let’s start again. What if you chose 10? What would she choose?

Does 15 beat 10?

No – 15 loses to 10 – look – on day twelve the player with 10 has already gathered 30 men – whereas the player who chose 15 nas nobody. 15 days loses.

Do any numbers beat 10?

There are some numbers that beat 10… 11 is one of them.

Is it possible for you to choose any number that is unbeatable?

No – There is no number that always wins. 10 beats all numbers less than 10 and all numbers more than 12. But 11 beats 10… and five beats 11…

Obviously the clan game is uninteresting. The second person to choose always wins the battle.

To help fix the bias – you put any number of numbers into a bagpipe. Your opponent then looks in the bagpipe and chooses any number. Finally, you play on the bag pipe until a number is spit out. That is your number. For example, you could choose to put the numbers 5, 10, 11 and 15 in the bagpipe.

We can organize the numbers in our bagpipe using arrows. The arrow from 5 to 11 means that 5 beats 11.

Your evil opponent looks in the bagpipe and chooses any number – she chooses six.

Six is going to beat 5, 10 and 15.

Six is going to lose to 11.

So the opponent will still win most of the time, but not all the time. She will win as long as the bagpipe doesn’t spit out the 11.

If she chose 12 instead of 6 – she would also win most of the time:

12 is going to beat 10, 11 and 15.

12 is going to lose to 5.

Your evil opponent could also choose a number that you chose.

For example, if you put the numbers 5,7,9,10,11,12 and 13 in the bagpipes – your evil opponent could choose 10.

10 loses to 11 and 12.

10 ties with 10.

10 wins against 5,7,9,13.

She still wins more than you ;-(

Your job is to find some numbers to put in the bagpipes so that your chance of winning is the same as your evil opponent.

Again – these five numbers don’t work because your opponent could choose 7 or 8 and end up winning more than you. ;-(

Experiment as a whole class. One person selects some numbers to put in the bagpipes – the rest of the class plays the role of the evil opponent and tries to find a number that will win most of the time.

For example – Bruce the Brave suggests putting 5, 10 and 11 in the bagpipes.

How can the rest of the class beat Bruce the Brave?

Stop and think.

Bruce the Brave is in trouble if the opponent chooses 6 or 12.

It is your turn. Try to find a group of numbers to put in the bagpipes that make your contest with your evil opponent equal.

Spoiler alert…

There is only one answer!

The only numbers that give you an equal chance against evil are 6, 8, 10, 11, 12.

Some battles are bigger than others. You could change the 30 objective to 10, 20, 50 or 100, but stop! To understand the problem deeply – your students should learn to think like mathematicians. That means trying small numbers like 1-10 to gain understanding. Not going for the nasty “100” right away.

Here is the solution for a bigger battle that is won when one side has 50 men more than the opponent. Your students may want to study it before tackling the smaller 1-10 battles.

Spoiler alert – I’m going to show you all solutions 1-100. I calculated all these by hand – helped by a student (Nicholas Lee.) Of course I could program a computer to do all this calculation super fast, but there is something meditative about revealing beauty one step at a time. This is poetry for me. I love the following slide!!!

This diagram can be read by choosing your battle size on the bottom X axis. For example – if you go to 30 you can read up that column that the numbers you should put in your bagpipe are 6, 8,10,11,12. We already knew that.

There is a lot of structure here. Your students should study it. What structure do they see???

Square numbers seem to be related to a single dot at the bottom… You can try to understand this later. Right now keep looking for patterns…

Another pattern is that the product of successive integers are somehow related to two dots at the bottom. So 5*6 = 30 and 6*7 = 42 and 7*8 = 56 …

There are empty regions that seem to follow a pattern…

The battles with target number 34-60 require seven numbers to be put in the bagpipes. 61-95 require nine numbers. The number of battles that require nine numbers is eight more. That is constant. The number requiring eleven numbers will be eight more again. Beautiful!

Now let’s take a step back and do some proofs.

Is it possible that an even number of numbers should ever be put in the bagpipes?

Spoiler alert…

No – an even number will never work because the evil opponent can always choose one of them that has more than half of its connecting lines being wins.

In this case the evil opponent can choose 10. It wins 4 times and loses only once.

Prove that any bagpipe solution must have an equal number of wins and losses for each number…

This is clear for the same reason. If a number like 10 wins against more than half of the other numbers in the bagpipe – the evil opponent will choose this number.

Hope this was as fun for your class as it was for me 😉