A Puzzling Square
Katie Steckles
I recently saw a video on YouTube, shared by fast-talking magician Pete Firman. It involves a puzzle with a square piece of wood, and a cloth bag – if you would like to watch it without spoiling the solution, you can pause when you hear him say “The secret to this puzzle is ingenious,” around 30 seconds in (after which he launches straight into a solution).
The trick involves trying to fit this square piece of wood into the fabric bag shown – and yes, the bag really is just a piece of fabric of the size shown, folded in half and stitched together at the ends, leaving a long side open.
The video states that you have to put the square inside the bag, “in such a way that all four sides of the square are covered by the bag” – along with the usual constraints of not being allowed to cut or destroy either of the objects in order to achieve this.
If you would like to go away and think about this now, please do. Then, before I discuss the solution, I would like to explain to you why I found this puzzle so interesting.
An Irrational Tangent
Back in January, I wrote about a set of coincidental recent discoveries that all involved a particular number, which was the titular Ubiquitous Surd known as the square root of three. But this is not my favourite surd, and there is another one which claims the crown, for many reasons.
The square root of two, a value of around 1.4142, has many enjoyable properties. As well as not being expressible as a fraction (and it being good fun to prove this fact in a variety of ways), it has many practical uses.
You may have seen Ben Sparks’ article back in August 2022, in which he explains that the A-ratio format of paper, used in most countries as the international standard, uses the ratio of 1:√2 to give paper a useful property – if you fold a sheet of A-ratio paper in half, the result will continue to be A-ratio, no matter how many times you do this.
This means we can easily copy two pages of a document onto a single page of the same size, and that the area (and thus the weight) of paper will neatly double if you increase the size to the next standard size up – making life easier for printers everywhere. But there is another cool fact hiding in a piece of A4 paper, which shows off even more of the cleverness of the square root of 2.
If you have a rectangular piece of paper and want to create a square one, there is a well-known method to make this happen: make a diagonal fold, taking one corner of the page and bringing it across until it touches the long side opposite. This fold will create a right-angled triangle which is exactly half of a square, and it will be sitting on top of a piece exactly the same shape, which is the other half of the same square – so cutting off the section to the right of it will leave a perfectly square piece of paper.
This square we have created from an A4 page has a fun secret. To see it, first note that if your original paper had a short side of length 1 unit, the long side, by our knowledge of paper ratios, would have been the square root of 2.
But if we consider the diagonal of this square – the length of that fold we just made – it is the diagonal of a square measuring one unit on each side. By Pythagoras’ theorem, we know this length will be the square root of 1² + 1², or the square root of two.
If you have a piece of A-ratio paper handy, you can try this – make the diagonal fold, and compare the length of this fold to the long edge of a piece of paper the same size: It should be exactly the same. This happens because the ratio we use for the paper, giving it all its wonderful halving-to-the-same-shape properties, is the same as the length of the diagonal – by definition. And if you have never noticed this pleasing coincidence, you can add it to your increasingly long list of reasons why you like the square root of two.
It’s in the Bag
Now you might be wondering why I am talking about this, given that I started off showing you a puzzle by a magician about putting a square in a bag. Well, when I first saw the video, and took a moment to think about how this might work, I noticed something interesting about the length of the rectangular bag relative to the side of the square.
The width of the bag obviously was not enough to fit the square inside; and the length of the bag, while it was longer than the side of the square, was not – for example – twice the length of the square, which might allow me to do something clever by folding the bag.
But there was an important and familiar ratio there: If the square of wood measured 10cm, overlaying lines on the video and measuring their ratios implies the bag is around 14.6cm long – suspiciously, just a shade over the square root of 2 times the side length of the wooden square.
This was a strong hint to me as to how the puzzle might be solved – I will not give it away, but if you want to go back and watch the rest of the video you will see how this comes into play. Sometimes this kind of mathematical insight is enough to point you in the right direction to a solution, and for me it was deeply satisfying to have used this realisation to figure it out.
Hopefully my obsession with the square root of two has become more understandable: This is only scratching the surface of the cool things you can do with it. If you need further confirmation that I love it maybe a little too much, you might be pleased to hear that 516 days (aka 1.414 years, rounded to the nearest day) after our wedding, my partner and I celebrated our √2 wedding anniversary by going out for triangular sandwiches – whose long diagonal side is approximately √2 times the length of the shorter side. Irrationally delicious!
The post A Puzzling Square originally appeared on the HLFF SciLogs blog.