[upbeat music]
[bell ringing over children chattering]
I love using puzzles and challenges as a way of making mathematics more engaging
and developing my students’ strategic thinking.
So today, I’m going to discuss a new approach to…
Hang on. That’s new? Hmm. Wait there’s a note. [cough]
Dear Mr. Greene. We know how much you love using puzzles and challenges
as a way of making mathematics more engaging and developing your students’ strategic thinking.
It’s true.
So, we’ve devised this magnificent mathematical mystery to test your thinking.
On the board you will see a collection of 10 squares.
Your task is to place the numbers 1 - 9 in each of these squares
so that each of the horizontal and vertical 3-square sections add up to the same total.
But we’re not going to tell you what this “target total” is.
At this point, you’re probably noticing that
we’ve only given you 9 numbers to place in 10 squares.
Well spotted!
To fix this, we’re going to allow you to use any one of the numbers 1 to 9 twice.
Oh yeah, we’re also not going to tell you what this “mystery number” is either.
From, your students.
Hmmm. Well this wasn’t how I was expecting to start the day.
So, let me get this right?
The task is to place the numbers 1 to 9 in the 10 boxes
so that these 2 rows and these 3 columns each have the same total.
And because we only have 9 numbers and 10 squares, I can use one of the numbers twice.
Sounds tricky?
Here’s what we’ll do.
How about you give this magnificent mathematical mystery a try
and check back in with me in a moment.
Good luck!
[playful music]
So, how did you go? Do you need a hint?
Well, I’ll tell you this. The target total for each row and column is 15.
Try again and see how you go!
[mysterious music]
This puzzle is actually a variation of a well-known
mathematical phenomenon called a ‘magic square’.
This is an example of a 3x3 magic square grid.
The goal is to place the numbers 1 -9 in each of the boxes
so that the total of the numbers in each row, each column, and both diagonals are the same.
This is what a completed 3x3 magic square looks like
and the sum of each row, column and diagonal is 15.
So why 15? For that we need a little algebra.
The first thing we need to find is the total of all the numbers we are allowed to use.
So that’s 1 + 2 + 3 + 4… all the way up to 9 which gives us a total of 45.
Now, let’s imagine each row in our 3x3 grid adds up to a total of m,
that is a number we currently don’t know the value of.
And we have 3 rows. That means that 3m is the total of all the numbers we can use.
Therefore 45 = 3m and m, our magic number, must be 15.
Now that works fine for a 3x3 magic square but what about the puzzle we are trying to solve?
Well, there are two main differences between a 3x3 magic square and our puzzle.
The first difference is that we don’t have to worry
about diagonals adding up to the target total
because we don’t have any three-square diagonals.
And the second difference is that we need to use one of the numbers twice.
The easiest way to visualise how this might work is to start with what's called a semimagic square,
that is a magic square where only the rows and columns
add up to the same total but not the diagonals.
This is one example of what that looks like.
Again, our target total is still 15.
Now it’s simply a matter of taking the centre box
and the box above it and sliding them down to form this new tail section.
The number in the middle of the top row becomes the repeated number
and the number in the centre box has been moved completely.
Of course, the position of these two numbers could be swapped without affecting the answer.
In fact, any of the numbers 1 through 9 could be the number used twice in this puzzle.
See if you can find solutions for all of them.
As a tip, it helps to write out all the available variations
of 3 number sums that add up to 15.
We hope you enjoyed today’s magnificent mathematical mystery.
Thanks for watching. We’ll see you soon.
[upbeat music]
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[upbeat music fades]