A Self-Enumerating Crossword

sallows crossword

Here’s a unique crossword puzzle by Lee Sallows. There are no clues — instead, each of the 12 entries must take the form [NUMBER](space)[LETTER](S), like so:

EIGHT BS
NINETEEN XS
ONE J

And so on. Can you complete the puzzle so that the finished grid presents an inventory of its own contents?

(A couple observations to get you started: Because the puzzle contains 12 entries, the solution will use only 12 letters. And one useful place to start is the shortest “down” entry, which is too short to be plural — it must be “ONE [LETTER]”.)

Click for Answer

A New Pangram

British recreational mathematician Lee Sallows has produced many varieties of the self-enumerating pangram — sentences that inventory their own contents:

This pangram contains four As, one B, two Cs, one D, thirty Es, six Fs, five Gs, seven Hs, eleven Is, one J, one K, two Ls, two Ms, eighteen Ns, fifteen Os, two Ps, one Q, five Rs, twenty-seven Ss, eighteen Ts, two Us, seven Vs, eight Ws, two Xs, three Ys, & one Z.

A few years ago he began to wonder whether it’s possible to produce a sentence that reckons its totals as percentages. This is more difficult, because the percentages won’t always work out to be integers. As he worked on the problem he mentioned it to a few others, among them British computer scientist Chris Patuzzo. And a few days ago, Patuzzo sent him this:

This sentence is dedicated to Lee Sallows and to within one decimal place four point five percent of the letters in this sentence are a’s, zero point one percent are b’s, four point three percent are c’s, zero point nine percent are d’s, twenty point one percent are e’s, one point five percent are f’s, zero point four percent are g’s, one point five percent are h’s, six point eight percent are i’s, zero point one percent are j’s, zero point one percent are k’s, one point one percent are l’s, zero point three percent are m’s, twelve point one percent are n’s, eight point one percent are o’s, seven point three percent are p’s, zero point one percent are q’s, nine point nine percent are r’s, five point six percent are s’s, nine point nine percent are t’s, zero point seven percent are u’s, one point four percent are v’s, zero point seven percent are w’s, zero point five percent are x’s, zero point three percent are y’s and one point six percent are z’s.

Details are here. The next challenge is a version where the percentages are accurate to two decimal places — Patuzzo is working on that now.

(Thanks, Lee.)

Sad Magic

sallows tragic square

The magic square at upper left arranges the numbers 3-11 so that each row, column, and long diagonal totals 21.

Lee Sallows found nine tragic words that vary in length from 3 to 11 letters and arranged them into the same square — and he found a unique shape for each word so that every triplet can be assembled into the same 3×7 shape, shown in the border.

Triplets

lee sallows triangle theorem

A pretty new theorem by Lee Sallows: Connect each vertex of a triangle to the midpoint of the opposite side, and place a hinge at that point. Now rotate the smaller triangles about these hinges and you’ll produce three congruent triangles.

If the original triangle is isosceles (or equilateral), then the three resulting triangles will be too.

The theorem appears in the December 2014 issue of Mathematics Magazine.

Stamps and Math

https://www.macaupost.gov.mo/Philately/XVersion/ProductList.aspx?admcode=MAC&emicode=201408&lang=en-us

Lee Sallows tells me that the postal system of Macau is releasing a new series of stamps based on magic squares. The full set will touch on everything from the Roman SATOR square to Dürer’s Melencolia. Details are here.

Charmingly, the values of the stamps will be 1, 2, …, 9 Macau patacas, so that the sheet of the nine stamps will itself form a classic Lo Shu magic square. Lee’s contribution, above, is a Nasik 2D geomagic square of order 3 — not only are all the rows and columns magic, but so are all six diagonals, including the four “broken” diagonals.

Somewhat related: In 2000 Finland issued seven stamps in classic tangram shapes, featuring images of science and education. (One of the small triangles, barely visible here, is a Sierpinski gasket.) Only three of the seven shapes are denominated postage, but I should think the temptation is overwhelming to arrange all seven on an envelope in the shape of a little man or a fish or something. I wonder what the post office makes of that.

http://philaquelymoi.blogspot.com/2014/06/stamps-with-interactive-games-update.html

Spanagrams

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

Self-Replicating Resistors

From Lee Sallows:

self-replicating resistors

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.

co-replicating resistors

(Thanks, Lee.)

Saving Face

http://commons.wikimedia.org/wiki/File:Geomagic_square_-_Joker.jpg
Image: Wikimedia Commons

“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):

sallows rep-tiles 1

The larger versions could then perform the same trick, and so on. Here’s another set:

sallows rep-tiles 2

In this set, each of the six pieces is paved by some four of them:

sallows rep-tiles 3

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.