Motorcycle Crankshafts

Why the 270 Degree “Crossplane Twin” is Suddenly So Popular
FortNine

Cool shots of crankshafts being forged. Nice, clear explanations of some obscure technical details. Great stuff all around.

Square Cross Puzzle

How did Ramanujan solve the STRAND puzzle?
Mathologer

I started watching this video the yesterday. I got as far as the square cross puzzle (timestamp 1:45 to 2:20), and that intrigued me. I stewed on it overnight and this morning I came up with this analysis.

Square Cross Problem

We start with a drawing of a cross. The cross is composed of five squares. The center of the cross is a square. Attached to each side of the central square is another square of the same size.

Our mission, should we choose to accept it, is to cut the cross into five pieces and then reassemble those pieces into two crosses with the same shape. These two new crosses are the same size as each other.

Since we are making two crosses from the original larger cross, each of the smaller crosses must be one half of the area of the original larger cross.

If the length of the side of one of the component squares of the larger cross is one, a little algebra will show us the the length of the side of one of the component squares of the smaller crosses must by the (2^0.5)/2 (the square root of two divided by two).

Cutting each of the component squares of the original square on both diagonals will give us 20 right isosceles triangles. Two of those triangles joined together along their hypotenuse will give us a square the same size as one of the component squares of the smaller crosses. Two triangles for each of five squares requires ten triangles to make one smaller cross. Two times ten is twenty, the number of triangles we got by cutting up the larger cross.

This is the brute force method. It shows we can cut up the larger cross into smaller pieces and reassemble them into two, smaller, crosses. However, we have twenty pieces, not five, so we do not have a solution to the stated problem.

The trick is to find cuts that do not need to be made so that when we cut up the larger square we only have five pieces. Now I'm thinking a computer program could make short work of this if I can just figure out how to encode it.

Cubic Trisection Puzzle

Cubic Trisection
You can easily make three identical pieces that can be put together to make a cube. Simply take six flat squares, group into three pairs and join the two squares of each pair at right angle. You can then assemble the three right angles into a cube. But they won't stay together. If you just set it down and don't touch it, it might hold together, but jostle it and it will collapse. This cubic trisection is a little peculiar. Make that very peculiar.

This puzzle was invented by Robert Reid. Doesn't seem to be much information about him, possibly since he spent 50 years running a cinema in Peru.

Via Dennis.

Teglon

Pumpkin

The Teglon is a fictional tiling problem from Anathem. Is akin to Penrose Tiles. The idea is that you have a set of some number of different tiles. The tiles are polygons with a bit of curved line painted on the face. The solution to this puzzle has to satisfy two requirements. The first is that the tiles must completely cover a regular decagon (a ten-sided polygon). The second is that the bits of line must all connect to form a single continuous line that leads from one vertex of the decagon to another. Like, I said this is a fictional puzzle, at least so far as we know, so it was nice to find this pumpkin where someone had engraved the solution. However, they left off the tiling pattern, so you will just have to work that out for yourself. Be forewarned, several historical people in this novel lost their minds trying to solve this puzzle.
     The web game Entanglement has some similar features.