Maths Games – Monopoly

When our children were small, we had a couple of games supposedly designed to help them learn maths. This really meant teaching them arithmetic and tables, which neither of them ever had any difficulty with anyway. And the games were DULL… They might have seemed fun in comparison to school activities, but they weren’t much of an attraction at home. So they didn’t get played often and long ago went to the jumble sale.

Any game involving dice helps with counting, but I particularly like playing Monopoly with kids. The rules aren’t too complicated, you have to add and subtract all kinds of numbers and work with money, you can play with three people or with lots, and almost everyone has some idea how to play. And, most importantly, it’s FUN. There’s enough going on to keep an adult interested, but it’s still worth playing if all you do is roll the dice and see what happens. Great for all ages from six (with a little help) to eighty six (or a little older), and a much better arithmetic workout than any of the so-called ‘educational’ games I know.

The main criticism that can be levelled at Monopoly is that it takes a long time. This is fair enough if you play to the last person standing, but is easily countered by having a house rule to end the game. When playing just with grown ups, we stop when the first player goes bust. With children, we usually use the dice from the ‘Lord of the Rings’ version on the standard British board and stop when the Ring reaches Mount Mayfair. (I don’t like the LotR board, or the way the Chance and Community Chest cards have been adapted. I’m not opposed to innovations – just to bad ones).

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The LotR dice. One of the 1s is replaced by the Ring symbol.  The Ring starts at ‘Go’ and moves along one property every time the Ring symbol is thrown. You could do the same thing yourself by having two different coloured dice and a rule like ‘Ring moves on red six’. I’m sure you can find a ring somewhere.

Monopoly also contains some nice probability lessons based on the distribution of scores obtained by adding two ordinary dice. There are 36 ways the dice can land [(1,1) (1,2) (2,1)…… all the way to (6,6)], and eleven possible scores from 2 to 12. The most likely score is seven which can be obtained in six ways [(1,6) (6,1) (2,5) (5,2) (4,3) (3,4)]. The next most likely are six and eight (five ways each), then five and nine (four ways each) and so on.

Now, consider which square on the board you are most likely to start your turn on. They are not equally likely, and here’s a picture to help you.

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Yes, it’s Jail. You can end your turn (and so start the next one) there by:

  • landing there

  • landing on ‘Go to Jail’

  • throwing three doubles

  • drawing a card

Over the course of a game you are more likely to start a turn by moving off this square than any other. And your most likely throw is a seven… which puts you on Community Chest on our board. But three of the four next most likely throws put you on the orange properties:

  • Bow Street needs a six (5 ways of getting this)

  • Marlborough Street needs an eight (also 5 ways)

  • Vine Street needs a nine (4 ways)

So 14 of the 36 ways the dice can land put you on an orange square if you start from Jail. That’s a better than 38% chance… and you’ll be leaving the Jail square more than any other. So as a first approximation, it looks like the orange properties are the ones to go for when you’re trading cards.

That’s not the whole story, and you will find lots more at sites like these:

http://www.retroactive-vintage-games.com/games-articles/best-monopoly-properties.asp

https://rijusarkar.wordpress.com/2013/09/04/probability-distribution-on-monopoly-squares-using-python/

Alternatively, you could get the family together, crack open a tin of sweets and have a game…!

Straight Lines Don’t Exist

No post for a couple of weeks ….. we’ve been busy painting the kitchen. And part of my mission was to get a straight line in the angle where the ceiling slopes down to the wall. The trick is to use masking tape, and take it off immediately after painting the line so the paint has no chance to bleed underneath. It looks all right, from a few feet away!

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While I was doing this (and between wondering why we didn’t just paint the whole thing white), I started pondering what a straight line is, anyway.

In one sense, we all know – it’s that thing you draw with a ruler and pencil for graph axes, triangles and so on. But if your school ruler is anything like mine was, it will get tapped on the edge of the desk, used for sword fighting and flicking paper pellets, wedged into gaps just to see if it will fit… The one thing it won’t have is a straight edge.

This is true of all straight lines we draw in the real world, however careful we are and however well we look after our equipment. At some level, visible or microscopic, they are not straight. They’re just straight enough for whatever it is we want them to do.

We also talk about going straight to somewhere, which usually means we take the most direct route (and don’t stop off for coffee and cake on the way). This makes sense: the shortest distance between two points is a straight line, isn’t it?

Well, yes, it is – on a plane surface (that is, a surface that does not curve in any direction). But that’s not much use to us Earthlings, inhabiting as we do the surface of a sphere (or a spheroid if you want to be pedantic about it). The shortest distance between two points on the surface of a sphere is a segment of a circle whose centre is the centre of the sphere (a Great Circle is the term used in connection with the Earth, even though it is not quite a sphere). This line is, of course, curved and is often referred to as a geodesic – a term more generally used in mathematics to mean the equivalent of a straight line in curved space.

And here’s a thought experiment. Imagine a spherical Earth, then draw a line due south from the north pole until you reach the equator. Turn through 90o and draw a line along the equator until you have gone 90o west. Then turn north again and draw a third line back to the north pole. This will make an angle of 90o with our first line. You’ve just created a triangle with three right angles – 270o! Who says all triangles have 180o? Curved surfaces are different!

If you want to know more about spherical geometry, there’s a lot on the web.  This is a good place to start:

http://euler.slu.edu/escher/index.php/Spherical_Geometry

And thinking of curves, if you are fond of calculus you might like to consider a straight line as a curve with zero curvature. In other words, the second derivative – the rate of change of gradient – is zero. And if the gradient is unchanging, the curve…. isn’t curved!

My head hurts. Here are some lines I made yesterday. They’re not straight, and it doesn’t matter in the least.

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