Magic Octagon

Can you predict where the arrow will point?

The front of the coin has an arrow pointing up. The back has an arrow pointing left. The two arrows are at right angles to each other. Simple enough.

Now turn the coin 45° clockwise and flip it around the vertical axis. Which way does the back arrow point?

It points in the opposite direction to the front arrow.

Reset, turn it 45° counter-clockwise instead, and flip it. Now the back arrow points in the same direction as the front arrow.

It isn’t a trick coin. It’s an ordinary octagonal prism with two arrows on it. The trick happens in your head.

Inspired by this Vsauce short. The original Magic Octagon coin comes from Raymond Hall of Physics Fun. This is my own 3D-printable model of the idea.

________________________________________________________________________________________________________________________________

How it work

Stop thinking “up” and “left”. Think in corners.

Each arrow is fixed to the coin and always points at the same corner, whatever you do with it. Looking at the front, the front arrow points at the top corner. The back arrow points at the corner two steps clockwise from it, the right-hand corner.

To predict the result, follow that second corner:

1. Turn the coin 45° and note where the back arrow’s corner ends up.

2. Flip the coin around the vertical axis. That corner jumps to the mirror position, from right to left.

3. The back arrow points at it.

45° clockwise: the corner moves to bottom-right, then flips to bottom-left, opposite to the front arrow.
45° counter-clockwise: the corner moves to top-right, then flips to top-left, the same as the front arrow.

The surprise comes from tracking the picture instead of the coin. The flip mirrors everything, so the back arrow seems to turn the opposite way to the front one.

________________________________________________________________________________________________________________________________

Print settings

Two variants:

Full coin

• Print it standing on an edge so both faces come out the same.

• Use a brim, because the footprint is narrow and the coin can tip over or come loose from the bed.

Halved coin

• Print both halves flat, no supports needed.

• Put the alignment pins into the corner holes and glue the halves together. Line them up carefully: the arrows must be at 90° to each other for the trick to work.

Infill and layer height don’t really matter. Use whatever your usual settings are.

________________________________________________________________________________________________________________________________

Source model
The original CAD is on Onshape: Magic Octagon on Onshape