Well having posted about the Wizzanator and a picture of a splayed pussy I thought I’d raise the tone a little with a bit of intellectual stimulus. This is a (very) early representation of something you probably know (or at least have heard of). What is it?

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Ok I’m game

and thick.

Could it be todays date in Cuniform writing?

1 1 was a racehorse 1 2 was 1 2 1 1 1 1 race, 1 2 1 1 2.

Just going back to my childhood here.

Sorry, but my ancient Sumerian is pretty rusty these days.

The opening lines of the epic of Gilgamesh maybe?

Some obscure pun that will make me want to commit violence on you when all is revealed?

Ok, its cuneiform numerals and top line goes 65, 72 , 96 I think

Second line duno, 120, dunno.

Can’t figure those stacked wedges at all.

Give up.

Is it 3.14159 und so veite?

JohnSF,

You’re getting there… The stacked wedges aren’t so bad. But strictly speaking you don’t need ‘em.

OK, folks:

Big clue. What’s special about 65,73 and 97 (it’s 97, not 96)? Not indivually but taken together.

NickM,

That should be 65, 72, and 97, I think. The squaw on the hippopotamus, and all that.

Yup, sorry typo PA!

PA nails it. They are all Pythagoreans triads like 3,4,5.

Oh and forget your squaws.

It’s the squire on the hippopotamus is the sum of the squires on the other two rides.

Great moments in the history of mathematics, eh?

It’s a good puzzle, and historically significant too. This is some of the earliest evidence around of people knowing about these triples. You can have a look at the original and see if you can figure out where Nick’s numbers appear.

Can you imagine, what it must have been like to be living back in the age of baked mud, and to already know these numbers? Weren’t they amazing? Is this an opportunity to have a good rant about the sad decline of educational standards?

Always one (or three) of the numbers is divisible by five. Is it obvious why?

It is to know the stars are there and to be trapped at the base of this gravity well. I shall look into the counter-puzzle. Cheers PA!

You set me thinking about the Minoans though… Flush, toilets, multi-story homes and bikinis 4000 years ago!

PA,

It isn’t obvious to me.

Well, obviously if 1 then 3 also follows as a possible because if a,b,c is a Pythagorean Triad then ka,kb,kc is too.

But why 5? in the first place. No idea.

NickM,

Five isn’t the only number I could have suggested, but when a long list of triads are written in base ten it’s more noticeable.

The way you do it is to know about arithmetic modulo 5. If you map every number on to its remainder after dividing by five, you get a nice little number system in which the rules of addition, subtraction, multiplication, and division all carry across perfectly. So the numbers …-8, -3, 2, 7, 12, 17, … are all represented by the mod5 number “2″. And If you multiply

anynumber from {…-3, 2, 7,… } withanynumber from {…-2, 3, 8,…} you always get a number from the set {…-4, 1, 6, …}. Each set acts as a new sort of “number”, that you can do arithmetic on consistently.So once you know this, and it’s a

brilliantlyuseful trick that has many more uses, the only other hard bit is to see that the only square numbers mod5 are -1, 0, and 1. And because there is this correspondence with proper multiplication, it means thatanysquare is going to leave remainder -1, 0, or 1. Squares that leave 2 or 3 are impossible.So for the equation a^2 + b^2 = c^2, it’s got to be entirely made up of -1′s, 0′s, and 1′s. You can have 1 + 0 = 1, and 1 + -1 = 0, and of course 0 + 0 = 0, but every way you do it always has to have a 0 in it, which means the real number that square comes from has to be exactly divisible by 5. If you try 1 + 1 or -1 + -1 on the left, you get a number which cannot possibly be a square.

That’s the mathematician’s way of doing it. There are other ways, but they’re horribly messy and you’re unlikely to figure it out without the mod5 map to guide you.

The idea of modulo numbers is very closely related to the idea of a number base. We use base 10, the Babylonians used base 60. A modulo arithmetic system is simply

the last digitof the number in some base. And it was the Babylonian invention of the place-value number system that made it all possible.So I suppose the answer to the puzzle is “no, it isn’t obvious,” any more than Pythagorean triads written in cuneiform is. But I think it’s quite interesting, nevertheless.

.

Now then, if I give you a whole number of many digits, ask you to raise it to the fifth power and tell me the last digit of the result, how fast could you answer?

Now you know the trick, this ought to be easy.

PA,

You’re still a double-plus cunt because this shit doesn’t occur to physicists. I now recall why I hate mathematicians. And I really do because I did 20 credits in discrete math (not

discreetmath). I therefore know modulo arithmetic rather well.You cunt PA, you utter cunt.

That’s a complement BTW.

Taken as such, Nick! Taken as such!

I kind of suspected you might, being in the computer trade, but I wanted to write for everyone who might happen by. I like to think that even people who know not very much maths might understand and learn a little. I can dream…

I’m glad you liked it, anyway.

(And by the way, my first degree was in theoretical physics, so I’m sort of a physicist too. I know what winds them up.)