Here’s an account of a hugely enjoyable school visit. By a happy coincidence, just two days later I was at a conference at London Olympia. A recent President of the Mathematical Association and I were both speaking about mathematical games. In his opening remarks he said Snakes and Ladders was of no interest as it offered no opportunity for decision-making. Fortunately I had some photographs of my visit with me, and I was able to tell him some of the things we’d done in school and encourage him to reconsider.
For a long time I’d had a large piece of cardboard about 120cm square in our garage. I knew it would come in useful one day, and when I got the opportunity to work with the Y4 class at Hawridge and Cholesbury School I knew just what I was going to do with it. I fetched it indoors and ruled it into six rows and six columns of 20cm squares and took it with me to the school’s “Numbers Day”.
Everyone knows Snakes and Ladders, so everyone was able to tell me that the square at the bottom left would be number 1, and the squares along the bottom row of the board would be numbered from 1 to 6, and in the next row up the numbers 7 to 12 would run from right to left, with the left to right / right to left pattern continuing.
This gave us plenty of chance to explore questions like “Where will 16 be?”, “What number is two squares above 18?”, and “What number is in the third square along in the top row?”
We discussed what similar questions might arise if I’d chosen to use a different number of squares in each row, but this was a brief digression. I’d also taken along some 20cm squares of coloured paper, and I invited the children to complete the board, which I found waiting for me when I returned an hour later. I’d also given some strips of various lengths for the children to make into snakes and ladders.
So the next stage was to position the snakes and the ladders. I’d rather assumed that we might choose to use a short, a medium, and a long ladder, and similarly with snakes. However, the class were adamant they wanted to use all five of each. For each, they considered and discussed the best placing. Their familiarity with the game ensured that in each case the positioning could be discussed with insight, rather than just slapping the snake / ladder down more or less randomly. Naturally we used blutack to position the snakes and the ladders rather than gluing them into permanent positions.
So we were now pretty well ready to actually try our game out. We agreed we all wanted to use the same simple rules (that you start off the board at position 0, there’s no need to throw a Six to start, and the first to reach or pass square 36 would be the winner).
Their decision to use all the snakes and all the ladders that we’d made, and their positioning of them, was triumphantly vindicated. Their ladder on 16 led to a snake on 22, which in turn sent them to a further ladder at 12, which invariably caused great delight. And the fact that our board had just 36 squares meant that games didn’t drag on – a typical game lasted under five minutes.
But I had one more trick up my sleeve, and this was the one that ensured our game became a high-quality investigative experience. I rolled the die three times, getting throws of 4, 6, and 1. If I use these throws in order I end up on square 5. But what If I can choose the order in which I use these three throws – can I get to 20 or even beyond?
So we now have an activity of almost limitless flexibility. I might give you four throws (say, 2, 3, 5, 5) to explore. Or I can ask you how far you can reach if I let you choose your own set of three throws. And what’s the smallest set of throws that will let you win the game?
And of course, as soon as we reposition any of our snakes and ladders we’ve got a whole new set of situations to explore.