Rock, paper, scissors....

I had some recollection of "bigger" versions of the game "Rock, Paper, Scissors". What I was really looking for was a way of making it more than two player. Unfortunately, a cursory search fails to uncover any means of doing this. I did find some other interesting stuff, though.

For example, it was used in the USA as a means of dispute resolution (I have never sounded more QI). A copy of the relevant court order, quoted in Wikipedia, can be found here ...

Upon consideration of the Motion – the latest in a series of Gordian knots that the parties have been unable to untangle without enlisting the assistance of the federal courts – it is ORDERED that said Motion is DENIED. Instead, the Court will fashion a new form of alternative dispute resolution, to wit: at 4:00 P.M. on Friday, June 30, 2006, counsel shall convene at a neutral site agreeable to both parties. If counsel cannot agree on a neutral site, they shall meet on the front steps of the Sam M. Gibbons U.S. Courthouse, 801 North Florida Ave., Tampa, Florida 33602. Each lawyer shall be entitled to be accompanied by one paralegal who shall act as an attendant and witness. At that time and location, counsel shall engage in one (1) game of "rock, paper, scissors." The winner of this engagement shall be entitled to select the location for the 30(b)(6) deposition to be held somewhere in Hillsborough County during the period July 11–12, 2006.

A five gesture variant is mentioned in The Big Bang Theory, apparently - "Rock, Paper, Scissors, Lizard, Spock". The existing relationships exist between the first three items, but then defines relationships between Lizard and Spock gestures and each of the other three. Details here. For the game to be balanced, each gesture has to win against half the other gestures and lose against the other half (which means that there should be an odd number of gestures).

But this has been taken even further by someone who went on to create 7, 9, 11, 15, 25 and eventually 101 (!!!!) gesture variants. Yes, you and your opponent can pick any one of 101 gestures, and the winner and loser is defined in each case. You can buy a poster - you'll probably need at least that.

Now, given no multiplayer version exists, is it possible to make one, I wonder?

Whilst we're on the subject of maths ...

Another thing that was never mentioned (as far as I can remember) was the interesting phenomenon in the multiplication square - you know, this thing ...

  X   1   2   3   4   5   6 ...
  1   1   2   3   4   5   6
  2   2   4   6   8  10  12
  3   3   6   9  12  15  18
  4   4   8  12  16  20  24
  5   5  10  15  20  25  30
  6   6  12  18  24  30  36

etc.

If you look down the diagonal axis from top left to bottom right, then you get a list of the square numbers - 1, 4, 9, 16, 25 ... What I noticed was that, if you go "northeast" and "southwest" from those numbers, you always get a number exactly one less. That is, if you take a number, and multiply the number one more and one less than it, then you get one less than the number squared. Or ...

(n - 1) (n + 1) = n² - 1

It turns out to be pretty trivial once you expand out the expression, of course ...

(n - 1) (n + 1) = n² - n + n - 1 = n² - 1

But nobody ever bothered to point it out, and I felt a gram of so of smug when I proved it for myself.

There's actually a more general thing lurking here ...

(n - k) (n + k) = n² - k²

... which means that if you look at the differences as you continue "northeast" and "southwest" from numbers on the diagonal, you are going to get another series of square numbers.

Happy Pythagoras Day

Not that this is a particularly well-known observance - actually, I just made it up, though it seems as though it did exist before.

We grown-ups have a kind of abiding folk memory of Pythagoras's Theorem - "For a right-angled triangle, the square on the hypotenuse is equal to the sum of the square on the other two sides." In studying Maths O-level and A-level, we had thrown at us over and over again triangles with sides having particular ratios. Most noticeably, 3:4:5, because

3² + 4² = 5²

Less commonly, we were also exposed to triangles with sides in the proportion 5:12:13 and 7:24:25, because

5² + 12² = 13²,

and

7² + 24² = 25²,

These are known as Pythagorean triples, and they form a fairly exclusive group. Conventionally (in the UK!) our shorthand for writing dates is dd/mm/yy. There are only two Pythagorean triples that in their lowest form, written from lowest to highest, encode a date - namely, 3/4/5 and 5/12/13. (Technically, 6/8/10 and 9/12/15 are also Pythagorean triples - but they don't really count, as they are multiples of 3/4/5). Thus, for the people who take nerdy notice of quirky numbers, 5/12/13 (ie. today!) is the last time we will see a date that is a Pythagorean triple for a long time. Hence Pythagoras Day.

I didn't have as much fun with this stuff as I might have done at school. (Yes, yes, I know that those of you who take pride in your mathematical ignorance will be appalled at the concept of maths being fun). I discovered for myself relatively recently that odd numbers form gaps between successive square numbers:

1 to 4 gap is 3
4 to 9 gap is 5
9 to 16 gap is 7
16 to 25 gap is 9

and so on. A series of Pythagorean triples can be built from this, as the squares of odd numbers are also odd numbers, and the gap between two square numbers is also an odd number, the sum of the two numbers:

The gap between 2² and 3² is 2 + 3
The gap between 3² and 4² is 3 + 4

So when the gap between two squares is equal to a square number, hey presto, you have a Pythagorean triple:

The gap between 4² and 5² is 9, which is 3²
The gap between 12² and 13² is 25, which is 5²
The gap between 24² and 25² is 49, which is 7²

They were the ones I knew about - but then I could see that 9:40:41 would be a Pythagorean triple, as would 11:60:61 and 13:84:85. Pretty neat.

However, Wikipedia takes the sense of achievement away by introducing Euclid's formula, which permits us to generate all Pythagorean triples. It's even more neat, but a bit soul-destroying. I just wish someone had shown me this stuff when I was at school!