CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC THE HINDU IN SCHOOL SUNDAY, NOVEMBER 12, 2017 10 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC NUMBER GAMES The best way to get a handle on this problem, in my opinion, is to try a bunch of examples. You might have noticed that parity, or evenness and oddness, has a role to play. DIMENSIONS OF THE TABLE CORNER THE BALL GOES IN Odd by Odd Top Right Odd by Even Bottom Right Even by Odd Top Left Even by Even Varies… The “even by even” case seems stranger, but it can be dealt with by a simple observation: a 2 by 6 table is identical to a 1 by 3 table (seen from a closer vantage point, if you like). In general, whenever you have a table with two even numbers as its dimensions, you can divide them both by 2 until at least one of them is no longer even, and you haven’t changed the nature of the table. We can predict, according to this pattern, where the ball will wind up in the big tables: 26 by 47 - Top Left 35 by 99 - Top Right 501 by 998 - Bottom Right 600 by 10,000 - Bottom Right, because 600 by 10,000 is really just a scale model of a 3 by 50 table. Now the real question: why does parity explain the path of the billiard balls? I want to offer two explanations, one elegant, one illuminating. First, the elegant explanation. For any table, draw it on a grid – or lattice, as we sometimes say – and colour the grid points in a checker-board pattern. Notice anything? The path the billiard ball takes can’t change from blue points to red points. That means we only need to figure out which of the four corners will be blue, or whatever colour matches the bottom left corner. The colour of the corners, as you can check, depends only on the parity of the dimensions of the table. So that’s the elegant idea, and it’s easy for it to go by quickly. Here’s another take. Imagine the ball rolling toward the end of the table, and instead of bouncing off, imagine it passing “through the looking glass,” into a mirror image of the existing table. In this case, using a 3 by 4 table, we can put 4 of them end to end vertically and 3 end to end horizontally. Colouring the corners of the original table to keep track, I can see that the mirror image straight-line path of the ball ends up in the bottom right corner. What matters here? Only the parity of how many tables I had to stack to the right and how many I had to stack up. Evenness and oddness, once again, explains everything. Then it bounces again, into a mirror image. Keep expanding the mirror images, and we can image our ball on a straight line path. Not only that, its journey through these mirror image tables is precisely a mirror image of its real path. In particular, if we keep track of the corner, we can still predict correctly where it will end up. There are some details to work out in both these arguments, and I encourage doing so. Parity is strangely common in mathematics and in mathematical puzzles, and it is worth getting to know, for its simplicity and its power.

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THE HINDU IN SCHOOLSUNDAY, NOVEMBER 12, 201710

NUMBER GAMESThe best way to get a handle on this problem, in my opinion, is to try a bunch of examples.

You might have noticed that parity, or evenness and oddness, has a role to play.

DIMENSIONS OF THE TABLE CORNER THE BALL GOES IN

Odd by Odd Top RightOdd by Even Bottom RightEven by Odd Top LeftEven by Even Varies…

The “even by even” case seems stranger, but itcan be dealt with by a simple observation: a2 by 6 table is identical to a 1 by 3 table(seen from a closer vantage point, if you like).In general, whenever you have a table withtwo even numbers as its dimensions, you candivide them both by 2 until at least one ofthem is no longer even, and you haven’tchanged the nature of the table.

We can predict, according to this pattern, where the ball will wind up in the big tables:26 by 47 - Top Left35 by 99 - Top Right501 by 998 - Bottom Right600 by 10,000 - Bottom Right, because 600 by 10,000 is really just a scale model of a 3 by 50 table.

Now the real question: why does parityexplain the path of the billiard balls? I want to o�er two explanations, oneelegant, one illuminating.

First, the elegantexplanation. For any table,draw it on a grid – orlattice, as we sometimes say– and colour the grid pointsin a checker-board pattern.

Notice anything? The paththe billiard ball takes can’tchange from blue points tored points. That means weonly need to figure out whichof the four corners will beblue, or whatever colourmatches the bottom leftcorner. The colour of thecorners, as you can check,depends only on the parity ofthe dimensions of the table.

So that’s the elegant idea,and it’s easy for it to goby quickly. Here’s anothertake. Imagine the ballrolling toward the end ofthe table, and instead ofbouncing o�, imagine itpassing “through thelooking glass,” into amirror image of theexisting table.

In this case, using a3 by 4 table, we canput 4 of them endto end verticallyand 3 end to endhorizontally.Colouring thecorners of theoriginal table tokeep track, I can seethat the mirrorimage straight-linepath of the ballends up in thebottom right corner.

What matters here? Only the parity of howmany tables I had to stack to the right andhow many I had to stack up. Evenness andoddness, once again, explains everything.

Then it bounces again, into amirror image. Keep expandingthe mirror images, and we canimage our ball on a straightline path. Not only that, itsjourney through these mirrorimage tables is precisely a mirrorimage of its real path. Inparticular, if we keep track of thecorner, we can still predictcorrectly where it will end up.

There are some details to work out in both thesearguments, and I encourage doing so. Parity isstrangely common in mathematics and inmathematical puzzles, and it is worth getting toknow, for its simplicity and its power.