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
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.


A self-reproducing sentence by Lee Sallows — “Doing what it tells you to do yields a replica of itself”:

sallows palingram

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.


Borromean Tribars

borromean tribars

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.

(Thanks, Lee.)

Pièce de Résistance

Take an ordinary magic square and replace its numbers with resistors of the same ohmic value. Now the set of resistors in each row, column, and diagonal will yield the same total resistance value when joined together end to end.

This paramagic square, by Lee Sallows, is similar — except that the resistors must be joined in parallel:

sallows paramagic square