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The charm of towers of Hanoi is that you can make a physical version where the legality of moves is easily verified because of the differently sized blocks.

I think the “you cannot have towers with the same amount of blocks” rule will make this a lot less charming, certainly if any of the numbers are larger than, say, 10.

There also is the issue of adding pegs, but that’s solvable by fixing the number of stacks (the triangular number for n is about ½n², so it certainly need not be larger than twice the square root of the number of blocks).

You can even make it smaller to get a variation on this game where the length of the list is limited.

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You can make a physical version by having one fixed height pole for each integer up to the max, and then an equal number of post holes. Eg. If you wanted to support a game up to the integer 10 you would need 10 posts with lengths from 1 to 10 units, and 10 post holes.

The biggest issue with this game is that there's no guarantee that any arbitrary starting state has a valid solution. A much needed improved would be a simple rule for guaranteed reversible starting formations (if one even exists).

Post holes are useless, because you can split left and then split back to the right. So all you need are the n tokens.

Hanoi isn't a good model for this, because the posts are not fixed in place and the units are all identical.

It's a bad physical game because the player has to do all the work to enforce the rules. The environment doesn't provide any useful assistance.

> If you wanted to support a game up to the integer 10 you would need 10 posts with lengths from 1 to 10 units, and 10 post holes.

You’d also need a way to represent the order of the posts (the game is about a list of integers, not a set, so you can’t move from [4,5,6] to [10,5], for example)

I think a halfway decent visualization is one where you have n different cylinders of lengths 1 through n and a gutter of length of the sum of the numbers you start with (in the [4,5,6] example that would be 15). Next, place the cylinders for the starting position in the gutter in the order given, so that it completely fills it. Keep the others elsewhere.

Allowed moves then are:

- replace two cylinders that are side by side in the gutter by one of the sum of their lengths that you have available.

- replace a cylinder in the gutter by two available cylinders that together have the same length.

I think this way to visualize the game also might lead to a physical construction, but I don’t see on yet.

I really like this. I could see constructing a physical artifact for a specific case of this game.

According to another comment, the best puzzle with a high number of 6 is [1,6,3] with a minimum of 14 moves.

This would mean you have 6 total rods, and two gutters. The puzzle gutter with a length of 10, and the storage gutter with a length of 11.

If you want more visual symmetry a high number of 7 allows you potentially 2 gutters of length 14.

Thinking it through a bit more, I would consider a variation on this game, where you don’t have a list but a ring buffer.

Then, you can replace the linear gutter by a circular one and replace the cylinders by parts of a torus. That would make for a cooler look of the game (on the other hand: how would you easily see you’ve completed the puzzle?)

Unfortunately, you won’t be able to put the ‘spare’ parts in a concentric circle as that would have a different radius.

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