A “self-interlocking” geomagic square by Lee Sallows. The 16 lettered pieces pave a single large square, and smaller squares can be produced by various groups of four pieces — those drawn from each row, column, and long diagonal, and 10 other symmetrically chosen quartets.
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In 1948 Melvin Wellman discovered this pretty anagram:
ELEVEN + TWO = TWELVE + ONE
And Dave Morice found this:
THIRTEEN + TWENTY – ONE = (NINETY / TWO) – TEN – THREE
Lee Sallows discovered two similar specimens in Spanish:
UNO + CATORCE = CUATRO + ONCE
DOS + TRECE = TRES + DOCE
These can be combined to make more:
UNO + DOS + TRECE + CATORCE = TRES + CUATRO + ONCE + DOCE
UNO + TRES + DOCE + CATORCE = DOS + CUATRO + ONCE + TRECE
From Lee Sallows:
In an electrical network, if resistors x and y are placed in series their total resistance is x + y; if they’re placed in parallel it’s 1/(1/x + 1/y).
This offers an intriguing opportunity for self-reference. Each of the networks above contains four resistors with values 1, 2, 3, and 4, and the total resistances of the networks themselves are 1, 2, 3, and 4. So any one of the numbered resistors in these networks can be replaced by one of the networks themselves.
The challenge was posed by Sallows and Stan Wagon as a Macalester College “problem of the week”; these examples were discovered by Brian Trial, an automotive electronics engineer from Ferndale, Mich. Sallows points out that any such solution has a dual that results from changing series connections to parallel, and vice versa, and then replacing all resistors values by their reciprocals.
This leads to a further idea: The two sets of resistors below are “co-replicating” — the four networks on the left can be used to replace the four resistors in any of the networks on the right, and vice versa.
“The Joker,” a picture-preserving geomagic square by Lee Sallows. The 16 pieces can be assembled in varying groups of 4 to produce the same picture in 16 different ways, without rotation or reflection.
The outline need not be a joker — it can take almost any shape.
Self-Tiling Tile Sets
These tiles have a remarkable property — by working together, the four can impersonate any one of their number (click to enlarge):
The larger versions could then perform the same trick, and so on. Here’s another set:
In this set, each of the six pieces is paved by some four of them:
By the fathomlessly imaginative Lee Sallows. There’s more in his article “More on Self-Tiling Tile Sets” in last month’s issue of Mathematics Magazine.
A self-reproducing sentence by Lee Sallows — “Doing what it tells you to do yields a replica of itself”:
This reminds me of a short short story by Fredric Brown:
Professor Jones had been working on time theory for many years.
“And I have found the key equation,” he told his daughter one day. “Time is a field. This machine I have made can manipulate, even reverse, that field.”
Pushing a button as he spoke, he said, “This should make time run backward run time make should this,” said he, spoke he as button a pushing.
“Field that, reverse even, manipulate can made have I machine this. Field a is time.” Day one daughter his told he, “Equation key the found have I and.”
Years many for theory time on working been had Jones Professor.
Nob’s Number Puzzle
A conundrum by the late brilliant Japanese puzzle maven Nob Yoshigahara.
Lee Sallows writes, “You have to solve this yourself, otherwise you won’t see how beautiful it is.”
Only the brilliantly inventive Lee Sallows would think of this. The figure above combines Penrose triangles with Borromean rings: Each of the triangles is an impossible object, and they’re united in a perplexing way — although the three are linked together, no two are linked.
A magic square by Lee Sallows. The 16 pieces progress in area from 1 to 16, and those in each row, column, and long diagonal can be assembled to form the same target shape with area 34.
Words and Numbers
In 1986 Lee Sallows invented the alphamagic square — spell out each number in this magic square and count its letters (25 -> TWENTY-FIVE -> 10), and you’ll produce another magic square:
David Brooks points out that this works also in Pig Latin.
Sallows extends the idea into geometry: