Is This Curve Infinite?

Hilbert from Abhishek Ruikar
This version in Oak

Out of the blue, a couple of months ago, I was contacted by Abhishek informing me of his latest creation and asking whether I wanted one. I had just mortgaged my soul to the devils (aka Dee Dixon and Tye Stahly) and asked whether he would wait a month for me to replenish my PayPal and also soothe over the disgruntled first wife (Whack! Ouch!)

He was very happy to wait for a bit and duly reminded me after a suitable period. Luckily, I had a little spare cash and it flew over the wires to India. This puzzle is available in Oak, Ash, Teak and Mahogany - I decided on the Oak version but they all look nice.

It drives in a nice green box (the corners of mine had taken a slight beating from the postal service) and inside is a little folded leaflet with the instructions (and if you need it, the solution).

Abhishek obviously has a penchant for knots and topology as his previous puzzle also involved wooden pieces with magnets that needed to be assembled into an intertwined shape. I needed to resort to Google to understand the name of the puzzle. I was aware of David Hilbert as a famous mathematician but did not know why this puzzle was named after him. Within a minute the reason sang out to me - the Hilbert curve is a fascinating concept describing fractal curves that can be both 2D and 3D space filling:

Hilbert curve courtesy of Tim Sauder

3D filling by Robert Dickau

When I removed the pieces, I realised that there were 8 of them and they all have similar but not identical shapes and now the reason for the name really became clear. This is a 3D filling puzzle:

8 very similar shapes with magnets!

I set to work making chains of pieces to try and form a cube. It didn't look too tough and was helped by the fact that the polarities of the magnets were the same on the equivalent ends of all the pieces. After about a ½ hour, I realised this was not quite as simple as expected. The pieces fit together making interesting shapes but after 5 or 6 of them the curve interested itself or blocked the insertion of the next piece. I made several dozen interesting shapes that didn't go anywhere:

Not getting anywhere!

Finally, I got fed up of random assembling of pieces and actually looked properly at the shapes that I had - there are 4 pairs of identical pieces which need to be arranged into the cube shape. It could not possibly be a random assembly - they had to work in a logical sequence which needed me to to some think©ing.

4 pairs of pieces

Once I understood this, then there was much less random trial and error. Thinking in terms of 6 faces and dividing up 8 pieces as 4 pairs was not helping me at all. It needed a few attempts at looking how the pairs could be arranged and thinking of edges (there are 8 in a cube) and I had a lovely little Aha! moment and managed to arrange my magnets in such a way that they all met end to end and formed a cube. Simply delightful - combining mathematics and mechanical puzzling cannot be beaten!

A Hilbert cube

Thank you, Abhishek, this was a delight. I have just realised that putting the pieces back in the box will be another challenge!

 I am sure that he would be delighted to sell you a copy if you contact him.