Funny mathematical puzzles.
All (almost, anyway) the material here is, totally without sense of moral, stolen from a book by Martin Gardner, Mathematical Puzzles & Diversions, and I have no claims of any kind...
Do you know what a Lo Shu is? It's a Chinese Magic Square, i e a matrix of numbers, organized in such a way that each row, each column and each diagonal add up to the same total. But this is not all with the Lo Shu. This particular magic square has a number of different qualities. They will all (?) soon be told here for your pleasure.
Hexaflexagons. A hexaflexagon is a hexagon that you can flex. An ordinary hexagon (or, in this terminology, a duahexaflexagon) has two sides. If you try to flex a duahexa, you'll soon find out that there are no other sides. But if you instead have a trihexaflexagon, you can first see one side, then turn it over, and see another. So far, it's exactly like the duahexa. But if you flex the trihexa, you'll find a third side. Amazing, isn't it!? And this is valid not only for a trihexa, but for a tetra-, penta-, hexa-, hepta-, octa- and enneahexa (with the only difference that they've got 4, 5, 6, 7, 8 & 9 sides respectively). In fact, one can quite easily make a dodecahexaflexagon. But hey, didn't I forget to tell you about the unahexa!? the unahexa is nothing else than a triangular Moebius-strip.
When you flex a hexaflexagon, you'll quite soon recognize a pattern in the different sides that turn up. The shortest way to "flex up" all sides is the Tuckerman traverse. I'll come to this some other time.
A complete mathematical theory of flexigations was worked out in 1940, by Tukey and Feynman (it has never been published, but parts of it has later been rediscovered by other mathematicians). They came up with a theory, that shows exactly how to construct a flexagon, any size and species.
Anyhow, the articles by Gardner in the Scientific American aroused many feelings, and he recieved hundreds of letters. These are two of them, published in the March and May issues in 1957.
SIRS:
I was quite taken with the article entitled "Flexagons" in your December issue. It took us only six or seven hours to paste the hexahexaflexagon together in the proper configuration. Since then it has been a source of continuing wonder. but we have a problem. This morning one of our fellows was sitting flexing the hexahexaflexagon idly when the tip of his necktie became caught in the folds. With each successive flex, more of his tie vanished into the flexagon. With the sixth flexing he disappeared entirely.
We have been flexing the thing madly, and can find no trace of him, but we have located a sixteenth configuration of the hexahexaflexagon.
Here is our question: Does his widow drag workmen's compensation for the duration of his absence, or can we have him declared legally dead immediately? We await your advice.
Neil Uptegrove
Allen B. Du Mont Laboratories, Inc.
Clifton, N.J.
SIRS:
The letter in the March issue of your magazine complainting of the disappearance of a fellow from the Allen B. Du Mont Laboratories "down" a hexahexaflexagon, has solved a mystery for us.
One day, while idly flexing our latest hexahexaflexagon, we were confounded to find that it was producing a strip of multicoloured material. Further flexing of the hexahexaflexagon finally disgorged a gum-chewing stranger.
Unfortunatly he was in a weak state and, owing to an apparent loss of memory, unable to give any account of how he came to be with us. His health has now been restored on our national diet of porridge, haggis and whisky, and he has become quite a pet around the department, answering to the name of Eccles.
Our problem is, should we now return him and, if so, by what method? Unfortunately Eccles now cringes at the very sight of a hexahexaflexagon and absolutely refuses to "flex".
Robert M. Hill
The Royal Institute of Science and Technology
Glasgow, Scotland
I got this letter from an Amanda Cox in Australia:
"Hi,
thought you might be interested in this puzzle..... I have been working on it for a few days now and its driving me nuts.
I was wondering does this particular puzzle have a name ?? I am really after the next line of the sequenced numbers.
1
11
21
1211
111221
Regards,
Amanda"
So I asked a friend of mine, Olle the Greatest, if he could solve it. (I haven't got the time!!! Or wits... ;)
His suggestion was this:
Pretty fast, you see that the sum of each row is
1
2
3
5
8
I e the Fibonacci series.
So, each new number should be created by the two last.
Every number ends with 1, so to be sure to maintain this, we move the last 1 from the second last number, and then put the remaining digits from the second number, and the last number, to the left of this digit.
From
a. 1211
b. 111221
We get
121 111221 1
(a) (b) (a)
Which sums up to 13, which is, of course, the next number in the Fibonacci series.
After this, we see if we can change 1s into 2s, which is done in pairs. We have
1211112211
we get (by replacing 1111 with 22)
12222211
But a 2 which is moved down becomes two 1s!! I e:
111222211
And that is the next number!!
Yesterday, I got yet another suggestion. It was Danny Moix, studying mathematics in Arkansas, who hade been thinking...=)
His solution went like this:
1, 11, 21, 1211, 111221, 312211
Each number describes the one before it:
1 One
11 One One
21 Two One(s)
1211 One Two and One One(s)
111221 One One, One Two, and Two One(s)
I propose the next line to be a description of 111221:
312211 Three One(s), Two Two(s), and One One
Following this method, the sequence would go like this:
1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221
Which, in Dannys eyes, is the correct solution (I write it like that as to provoke a conflict... ;)
So, I thought, why don't arrange some kind of contest? You vote for the solution that's the most appealing to you, and if you dont' find a solutino like that, you come up with your own one!!
It'll have to start with e-mail, but who knows, maybe I'll make some kinda cgi script...
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