Possible Puzzle Modes

Easy Puzzle (Fun, tactile, relaxing)

Get as many 2x2 solves as possible.

Medium Puzzle (A little more challenging)

Get as many 3x3 solves as possible. How many are there?

ULTRA Puzzle (Nearly Impossible)

Get the 4x4 solve! There is only one!

COMBINED ULTRA

Combine corner pieces from the series of 4 ULTRA puzzle solutions (4 corners for each of 4 puzzles = 16 pieces).

MEK ULTRA Circle (https://makerworld.com/en/models/374776#profileId-274729)

MEK ULTRA Square (https://makerworld.com/en/models/559020#profileId-478205)

MEK ULTRA Triangle (https://makerworld.com/en/models/621485#profileId-545546)

MEK ULTRA Hexagon (https://makerworld.com/en/models/623400#profileId-547626)

MEK ULTRA Combined Holder (https://makerworld.com/en/models/623474#profileId-547697)

Play

Each piece has two tabs and two slots for interlocking or connecting pieces. I like to call them innies and outies, but there are many terms for such a key/lock type connection (tongue/groove, loop/socket, tab/blank, knobs/holes,...).

Each connection needs to interlock, so two ‘slots’ can’t border and abut each other in order to correctly ‘solve’ the puzzle - even though it is actually physically possible to fit them side by side 🙂. Two tabs next to each other is a little less physically possible, but still not valid.

Pieces can individually be oriented in 4 ways per side for a total of 8 possible orientations after being flipped.

There are many possibilities to find custom solutions (2x2, 3x3, 3x2…), but the single ULTRA solution is 4x4 pieces with all edges interconnecting.

Solve the COMBINED ULTRA and win a prize!

I will likely give the first person a reasonable prize, but after that the prize will probably drop to me saying CONGRATULATIONS and trying to find an opportunity to work with your genius or try to get you to come work with us at the Mayo Clinic 🙂. But solve it FIRST and let me know!

How to describe your solutions - Puzzle Solve Notation

The pieces have arbitrary ‘top’ and ‘bottom’ designations designated by whether the direction arrow comes before the number ('bottom') or after the number ('top'). The pieces are randomly numbered 01-16 with two digits. When recording solutions, the direction arrow can be used to both indicate the piece’s directional orientation in your solution as well as whether it is ‘top’ or ‘bottom’ facing up.

Below are some recording/notation examples:

2x2 solve

The picture above could be described as follows:

06↓, 09→, →02, ↑08

The arrow after the 06 indicates it is the ‘top’ face, the arrow pointing down indicates the directional orientation of piece 06. Each piece follows going from top left to bottom right.

Same 2x2 solve rotated clockwise:

↓02, 06←, →08, 09↓

Same 2x2 solve flipped over:

→06, 02↓, ↓09, 08←

Same 2x2 solve using ”standard notation” (lowest number piece in upper left corner and its orientation being ‘top’ facing - so the arrow to the right of number):

02→, 08↓, ↑06, →09

MEK ULTRA COMBINED puzzle

Notations for a solution would include the MEK ULTRA Puzzle's edition as well as number and orientation since 4 corner pieces from each edition will combine for this puzzle.

The standard notation would count first edition numbers (Circle) lower than subsequent editions when looking for the lowest number corner piece to put in the upper left corner with its ‘top’ facing up.

Circle < Square < Triangle < Hexagon

example: (Circle-↓012), (Square-02←), (Triangle-01↑),…

Some notes on Numbers and Combinations

Brute Force - don’t brute force it

Because of the flipping possibility we can modify the formula from https://math.stackexchange.com/questions/270873/combinations-of-a-jigsaw:

Replacing the 4 with an 8, as each piece can be in 8 orientations.

C(n) = n! * 8^n

16! * 8^16 = 5,889,241,796,446,748,858,646,528,000 = 6e21 = 6 sextillion

Each one of those 4x4 combinations would actually have 7 combinations that are actually the same combination just the whole 4x4 rotated and/or flipped - so we can divide by 8 🙂, so less than a sextillion, pretty much nothing.

Brute force, but ‘pruned’

If you traverse each tree of combinations, but stop if the next piece doesn’t ‘fit’... So, for example, you lay the first piece down in position1 and orientation1. Then use the second piece at position2 going through all 8 of it’s orientations and none of them ‘fit’, then you wouldn’t need to go through all future pieces and orientations in the tree for positions 3 and forward. If you do this type of brute force algorithm then for this particular puzzle you would need to check 2,087,387,360 = 2e9 = 2 billion.

Again that would get you all the combination checks to find all 8 orientations of the one ULTRA solution. If you did these checks one per second it might take 60+ years, but if you stopped once you found one of the orientations then it would only take maybe 3 or 4 years. That’s totally doable!!!

Using More Logic

The fun is in developing reasoning around how to solve it, so I won’t give away any of those possibilities, but it is a challenge even after establishing rules to prune the solution space.

Background and History

MEK ULTRA!? Is this part of some weird human experiment (https://en.wikipedia.org/wiki/MKUltra)? No… or is it…

No it’s not - this was inspired by the concept of - The Ultimate Puzzle 48 puzzles rolled into 1 by Lee Willcott.

When solving ‘The Ultimate Puzzle’, we made a bit of a mistake. My son solved it while we were messing around with it on a family vacation and I thought ‘that’s weird, I thought it would be harder'. Well, the place we were staying on vacation didn't have great lighting at the table where we were doing the puzzle. We didn't recognize that there was a front and back side to the pieces (the front was a little more glossy). And the intent was that you solve it with all the pieces facing up. But the joining parts of the pieces (tabs/slots) were symmetrical and you could connect pieces regardless of whether they were oriented face up or not.

As I sometimes do, I wrote some software to analyze the heck out of the game. It was, in fact, much easier to solve (there were thousands of possible good solutions) when allowing the pieces to be glossy side up and down. The side of the box says 250K incorrect solutions - well if you allow using the pieces oriented up and down then there's a heck of a lot more incorrect solutions, but also there are more correct solutions. In fact, it's much easier to solve the 4x4 if you allow up and down orientations. Rather than the real 48 solutions, there are thousands.

I liked the idea of a puzzle allowing up and down, but wanted it to still be very difficult. So I went on to attempt to generate a single solution version of a new puzzle allowing up and down orientations of pieces. There is an exponential increase in incorrect answers, but I needed to narrow down the solution to one. So it took some work.

Another thing I liked about the Ultimate puzzle was that the inward and outward arrows would sometimes get you confused - you would get all excited for a moment when you're getting close to solving and - nope, not the arrow you were looking for. So, why not make the other two tabs/slots something similarly hard to distinguish as well - I went with the top and bottom of a 5 point star. Embrace the frustration!

Finally, when generating the MEK ULTRA puzzle I wanted to add another dimension. I was able to generate several sets of 1 in a billion solution puzzles. Then I was able to generate a MEK ULTRA COMBINED puzzle from corners of 4 sets that would be another 1 in a billion. ENJOY!