Move a match
How can you arrange these 10 matches in four piles so that when you
move one match from three of the piles into the fourth, you end up
with the same arrangement?
Age
7 to 11
Challenge level
Problem
Take $10$ matchsticks and put them into four piles.
For example, you could have a pile of $1$, two piles of $2$ and a pile of $5$:
Image
Now, the idea is to take one matchstick from each of three of the piles and add it to the fourth.
So, for example, we could take one stick from the two piles of $2$ and one from the pile of $5$ and add them to the pile of $1$. This would give:
Image
So now we have two piles of $1$ and two piles of $4$.
We have gone from piles of $2, 2, 1, 5$ to piles of $1, 1, 4, 4$.
How could you put the $10$ matches in four piles so that when you move one match from three of the piles into the fourth, you end up with exactly the same distribution of matches in four piles as when you started? (In other words, in the case above we would have gone from $2, 2, 1, 5$ to $1, 5, 2, 2,$ for example - although this isn't possible!)
What can you say about this starting arrangement of the matches?
Could you arrange, for example, $14$ matchsticks in such a way as to be able to make it work too?
How about $18$ matches?
In general, what can you say about the way in which you must arrange the matchsticks for this to be possible?
Getting Started
How could you arrange 10 matchsticks in four piles?
Think about whether you might need piles which are the same or different to start with.
If you have a pile with just one matchstick at the beginning, what must happen to that pile?
Student Solutions
Congratulations to everyone who helped crack this Toughnut, and correctly spotted, as Samantha has said:
"In order for this solution to be possible, the number of matches in each of the four groups must be consecutive."
So aj explained for 10 matches:
"for 10 matches, you can arrange them as 4,3,2,1. Take a match from each pile from the left and add them to the pile at the end. You end up with 3,2,1,4."
Congratulations to everyone who noticed that in general you take from all but the smallest pile, and place the matches on the smallest pile, as Jake and Hannah (St Peter's School,Wrecclesham):
Image
Teachers' Resources
Move a Match
Take $10$ matchsticks and put them into four piles.
For example, you could have a pile of $1$, two piles of $2$ and a pile of $5$:
Image
Now, the idea is to take one matchstick from each of three of the piles and add it to the fourth.
So, for example, we could take one stick from the two piles of $2$ and one from the pile of $5$ and add them to the pile of $1$. This would give:
Image
So now we have two piles of $1$ and two piles of $4$.
We have gone from piles of $2, 2, 1, 5$ to piles of $1, 1, 4, 4$.
How could you put the $10$ matches in four piles so that when you move one match from three of the piles into the fourth, you end up with exactly the same distribution of matches in four piles as when you started? (In other words, in the case above we would have gone from $2, 2, 1, 5$ to $1, 5, 2, 2,$ for example - although this isn't possible!)
What can you say about this starting arrangement of the matches?
Could you arrange, for example, $14$ matchsticks in such a way as to be able to make it work too?
How about $18$ matches?
In general, what can you say about the way in which you must arrange the matchsticks for this to be possible?
This problem will challenge pupils to compare numbers, think ahead and reason logically. They might also be visualising the consequences of moving particular matches, although ideally, there would be real headless matches or lolly sticks available for them to try out their ideas.
In order to encourage children to think logically about this problem rather than just trying random arrangements, you could use the suggested hints as questions. Pupils could start off by working through the possible arrangements of the 10 matches in a systematic way, for example by beginning with three piles the same and one different (3, 3, 3, 1 or 2, 2, 2, 4 or 1, 1, 1, 7) then two pairs of piles the same (4, 4, 1, 1 or 3, 3, 2, 2) then two the same and two different (2, 2, 1, 5 or 1, 1, 2, 6 or 1, 1, 3, 5) then all different ... (Finding all these combinations is a challenge in itself.) Chances are that, as they do this, they will begin to realise certain things about whether piles can be the same as each other or not and what happens if there is a pile of just 1 match. However they discover that consecutive numbers of matches isthe key, there is still more to be investigated if different numbers of matches are used which can then lead to a complete generalisation.