The Search For Ultimate Snakes and Ladders

For nearly 35 years I’ve run my own games magazine, originally by post and now of course by email.  Some really expert gamesplayers have taken part, and one game we played with great success was, believe it or not, Snakes and Ladders.

Here’s an introductory version we’ll call Invisible Snakes, and Ladders.  As the Gamesmaster, I’ll publish a board showing only the Ladders.  I am the only person who knows the positions of the snakes, but I will tell you they’re not positioned randomly (for example, their heads may be on multiples of 11 with each snake sending you back 6 spaces).  Furthermore, each player may choose their own die-roll each turn, with the proviso that in your first six turns you must use each number 1,2,3,4,5,6 exactly once.  For the next set of turns you again have to use each number once – this is easily managed by giving each player a set of 1 to 6 cards.

So as the game goes on players can deduce something about the nature of the snakes and plan how to make the most of their knowledge by using their die rolls to the best effect.  It’s a game full of observation, hypothesising, and strategy.

Of course, I can make things a little more interesting by making the Ladders invisible as well as the Snakes.

And when you play with really expert gamesplayers they’ll want things to be as challenging as possible.  So if you’ve got invisible snakes and invisible ladders, the next step is to make the board invisible as well!

So now we really do have something close to Ultimate Snakes and Ladders.  As well as having to deduce the positioning and effects of the snakes and the ladders, players have to find how the board is laid out.  Is it a normal S&L board or a conventional Hundred Square?  Does the numbering run from left to right or in another direction?  Is the board in ten columns or six, or eight, or eleven?  In fact, is the board rectangular at all (I recall we used triangular layouts and perhaps even circular ones)?

I recently heard an eminent maths educator say Snakes and Ladders held no interest because it was purely random, so that’s why I’ve done this series of four posts.  And I haven’t even mentioned shrinking ladders (which wear down as they’re used), or growing snakes (which expand as they feed), let alone boards with Trap Doors!

Our new National Curriculum in England demands that mathematics should incorporate Fluency, Reasoning, and Problem-Solving, and there’s no reason why Snakes and Ladders shouldn’t stimulate all three of these aims.

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Yet More Snakes and Ladders – Multiple Patterns

We’re used to the regular patterns multiples display on a conventional Hundred Square.   Multiples of 2, for example, fill the even-numbered columns, and you can probably visualise without much difficulty the different, but equally regular pattern they display on a Snakes and Ladders Square.

As a mental exercise, try a couple more.  Start with multiples of 10, and that should give you a start to see if you can predict what happens with multiples of 5.

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More Snakes and Ladders – The Grid

One of the reasons I’m fascinated by using Snakes and Ladders is because the board offers so many interesting comparisons with the conventional Hundred Square.

We’re all familiar with the standard 1-100 Hundred Square, and to make life easier I’m going to orient it with 1 at the bottom left and 10 at the bottom right – just like a Snakes and Ladders board in fact.

There are lots of properties of the standard Hundred Square and we probably take many of them for granted:

The numbers show a regular and predictable pattern.

The number 1 is in the bottom left-hand corner and the number 100 is in the top right-hand corner.

Each of the numbers from 1 to 100 occurs exactly once in the square.

You can locate any number (e.g. 42, 76) on a blank square without having to work out the positions of all the other numbers in the square.

The numbers in each row increase as you move from left to right.

A typical row goes from 21 to 29, and ends on 30 – the first number ends in 1 and the last number ends in 0.

As you move up a column the numbers increase by 10 each time.

The numbers in a column are either all odd, or all even.

The diagonal from bottom left to top right shows the multiples of 11.

Multiples of 2 show a regular pattern. Similar patterns are shown by all the other multiples.

It’s well worth discussing each one of these. Some are preserved in a Snakes and Ladders square, but others are modified.

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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.

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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.

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Traffic Lights

I invented Traffic Lights because I wanted a five-minute game with the accessibility of Noughts and Crosses but also enough depth to make players think a bit. And whereas Noughts and Crosses is frankly a bit boring once players have discovered the thing to do is grab the central square and then with sensible play you must get at least a draw, in Traffic Lights draws are impossible and someone has to win.

Here are the rules:

1……The game is played on a 3×3 board of squares. At the start of the game all the squares are empty.

2……When it is your turn to play you have three options:

• (a) you may place a red counter into any empty square, or
• (b) you may replace an existing red counter with an orange counter, or
• (c) you may replace an existing orange counter with a green counter.

3……You win if you place the final counter that completes a row / column / diagonal of three pieces of the same colour. (Whoever played the previous counters in the row is immaterial; it is the third counter that wins the game.)

Here’s the start of a game.  I play first:

And I play a further red:

Your turn.  Time to introduce an orange (by the way, I found my orange counters came out looking very reddy, so I’m using yellow ones).

There’s a nice sense of momentum in the game.  The board begins to fill and the number of possible choices decreases, while the number that lead to instant disaster rises.  After a few turns we’ve reached this stage, and there only seven possibilities, of which – I think – four lose immediately:

For example, promoting the red at the bottom left gives:

And this allows the promotion of the last red, creating a winning orange line:

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Traffic Lights has always been just about the most popular of my games.  I think it probably first appeared on NRICH at  http://nrich.maths.org/1181 .  It’s been played by six and seven yearolds as well as adults, and has often been used in competitions for primary pupils in Portugal.

Once you’re used to the basic version, you ought to play on a 3×4 board.

There are some more references at  http://www.di.fc.ul.pt/~jpn/cnjm8/

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What’s Your Game? is a book I co-authored; it was an attempt to demonstrate how games could be both interesting and worthwhile to play, and also stimulate investigative learning in Key Stage 3 maths classes.  Most used only the simplest of equipment and for each  we included worthwhile mathematical questions.  Several were accompanied by copies of interesting work from pupils.

I was never more than partially satisfied with it, but since my contribution was largely restricted to supplying the games while Mike Cornelius did all the hard work I figured I didn’t have much right to complain.  There were getting on for fifty games – Nimble (see October 2013) was one, and I’ll probably write about a Noughts and Crosses variant called Traffic Lights sometime.

It good some decent reviews but it appeared at a time when schools were nervous about anything that didn’t have National Curriculum on the cover in large letters, and it didn’t sell many copies.  I still get a royalty cheque for £0.00 from Cambridge University Press regularly even though it’s been out of print for a good while.

However, you can of course find copies on Amazon.  I was delighted to find that one seller values it at £2919.68, though another stockist only wants a measly £303.45 .  More realistically there are currently three sellers who’re asking £3.00 .

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There Are Polar Bears Round The Icehole

It’s good to know that something you’ve done has stuck with someone; out of the blue I had a call from someone I hadn’t seen for ten years or more.  “Hello Alan, can you remind me about the Polar Bears?”

So here’s the activity that Gaynor wanted to talk about.

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I roll five dice:

“There are six polar bears round the ice-hole.”

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Again:

“There are two polar bears round the ice-hole.”

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Once more:

“There are no polar bears round the ice-hole.”

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It’s much the best fun to do this informally when you can test out all sorts of ideas, but I’ll have to tell you that the colours of the dice don’t matter, nor their size, nor their positions.  (Generally speaking, I’ll use five dice, but we could use a different number.)  We’re left with just one factor to consider, that it’s the numbers showing on the top faces of the dice.

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Here are a few more:

“There are four polar bears round the ice-hole.”

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“There are no polar bears round the ice-hole.”

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“There are eight polar bears round the ice-hole.”

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“There are ten polar bears round the ice-hole.”

Any ideas?

Nimble

I don’t think anyone can beat this as a two-player mathematical game that’s so easy to use and so rich in development.

Write down the numbers from 1 to 9.

When it is your turn to play you MUST cross out the highest number in the line and you MAY cross out the next highest as well.

So here’s my first play (I’m red):

And here’s my opponent’s:

And my next play:

Play continues like this until one player wins by crossing out number 1.

Green wins!  Curses – I must do better next time.

Chess In Schools and Communities

Recently I was invited by Chess In Schools and Communities to talk to them about National Curriculum mathematics and I spent a delightful day with an inspiring group of people.  CSC have a project in which a chess tutor works alongside the Y5 teacher for a lesson a week; their task is to teach the children chess and the project will be evaluated to see if there is an effect on the children’s mathematics learning.  A separate project at Manchester Metropolitan University has already found some very positive suggestions that Y3 chess-players did substantially better when given a test involving non-verbal reasoning, maths, and problem-solving questions.

There are some areas of the Y5 programme of study where mathematics can clearly draw benefits from chess – co-ordinates, symmetry, translation.  Even more obvious are the links with the problem-solving thinking we’re accustomed to call Using And Applying Mathematics, where the current Y5 curriculum requires children to “Solve problems …, Explore puzzles, find / confirm solutions …, Plan and pursue an enquiry …, Explore patterns and relationships, …, Explain reasoning, …”.

I’ve spent a lifetime using games in the classroom, and I suggested that using games offers four main benefits.  A game like chess contributes to all four, and the maths classroom will draw particular benefits from the third and the fourth:

…… Games have a socialising value – children need to take turns, win and lose with grace, abide by the rules and the etiquette,

……Games are part of our cultural heritage (chess, cards, dominoes, backgammon, …); and from outside the UK chess itself, and Go, mancala, …. ,

……Games give enjoyment, and an environment in which children can accept high levels of challenge,

……Playing a game requires skills of observation, analysing, forming a plan, asking “What happens if?”  These have a very high correlation with the skills used in Using and Applying Mathematics.

You can find out about CSC at  http://www.chessinschools.co.uk/

http://educationendowmentfoundation.org.uk/projects  will give you information about the research project.

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End Of The Line

Children love games because they’re fun; I love them because they’re fun and because they tell me so much.  I learn the methods children are comfortable with, and because I can ratchet up the demand so they cheerfully accept challenges that they wouldn’t dream of handling via a worksheet.  And games stimulate children’s thinking in hypothesising, in problem-solving, and thinking ahead.

Here’s a very simple game that demands children simply have to call upon these thinking skills.

You need a set of cards, numbered from 1 to 12.

Mix them up and deal them out, face up, in a line.  When it is your turn to play you can choose to pick up either of the two cards at the end of the line.  Players take turns to do this, and the winner is whoever collects cards with the greater value.

The obvious play is to pick up the 10, but that lets your opponent choose the 12.   Perhaps it’s better to start with the 6; then whichever card your opponent takes then you get either the 11 or the 12 next turn ….

The game is very easy to use, and there are lots of ways of developing it – it works with three players, and you can use cards with different sets of numbers – negative numbers, fractions, ….

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