*My wife would probably allege that if the house were on fire my first priority would be to make sure the paper trimmer was safe. Not quite true perhaps, but it’s certainly true I do have a bit of an addiction problem. My cosy office is less than about 3 metres square, but it does hold five paper trimmers.*

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I make a lot of cards for number and sorting games for my pupils, and the other day I needed to convert a 24cm x 24cm card square from a cereal packet into 36 4cm x 4cm cards.

It seems intuitively obvious that some ways of cutting up the square will be more efficient than others, but in spite of my long experience I wasn’t clear what the best strategy would be.

Would it be best to (i) first slice the 24cm x 24cm square into six strips, or (ii) first cut the big square into quarters, or (iii) perhaps cut the big square into six 8cm x 8cm squares and then cut these up? Or perhaps another strategy is better than any of these.

Over to you. If you’re going to put yourself into my shoes I need to emphasise that I’m cutting card rather than paper, and my trimmer cannot handle more than one thickness. Do let me know how you get on with this exploration, which I’ve never seen offered anywhere else.

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I’ve been down this route before and I’ve discovered that the best way to make the fewest cuts is to minimize mistakes. Really.

There’s an excellent strategy that I recommend. First, DO NOT start off by cutting the paper into 4 cm strips. A margin of error will catch up with you by the last cut and you will find your final squares to be a bit unsquarish. Instead, start off by cutting 8 cm strips. You will have three of them. Then reduce your strips to 8 cm squares. Finally, cut these in half, both ways, and you will have a delightful pile of squares.

This afternoon my husband came home with a huge cutting mat. He had seen it at a tag sale for $16.00 . These sell for over a hundred dollars. It fills up a full quarter of a full sized ping-pong table. Now I am sure that he really loves me.

If you find a more efficient way to cut, let me know! Good luck.

So your strategy requires a total of 35 cuts – the crunch question is how many would be needed if anyone did the initial slicing into six separate strips?

(You’re quite right to point out that in practice it’s hard to avoid producing some squares that wouldn’t pass a quality control inspection, but I’m assuming here we’re all skilled enough to generate perfect squares every time.)

Starting with 6 strips is still 35 cuts…5 cuts for the 6 strips, then each strip would need 5 cuts, right? Even after cutting paper for years, I still would never start with 6 cut strips…the tiniest errors magnify…It’s really interesting for me to read about you wondering aloud about this cutting question, as it’s one that I ask myself on a fairly regular basis, when I am prepping for the bookmaking workshops that I teach. I don’t think I have ever found a way to make less cuts, no matter how I slice it. I’m curious to know what you arrive at after examining this!

You start with 1 piece of card, and each cut gives you one extra piece of card becuase you can only cut one thickness at a time. So you will always need 35 cuts.

Perfect! However carefully I try and minimise the number of cuts there will always be 35 needed. It’s reminiscent of the tennis tournament problem – if you’ve got 20 players in a tournament you’ve got to eliminate 19 of them, so a total of 20 matches will be needed.

By the way, I did post the problem on 1st April!