3Y + 1W Conjecture – Collatz

A puzzle being solvable is only conjecture until it can be proved that it is possible.  Some folks use fancy shmancy mathematical proofs while the rest of us just bash our heads against it until it yields.  Or doesn’t, in which case, Q.E.D., it is relegated to the obviously unsolvable pile.

On the mathematically unsolvable pile lies the Collatz Conjecture.  The objective of this puzzle by German mathematician Lothar Collatz is to prove that any positive integer can be reduced to 1 in a finite number of steps where at each step the number is divided by 2 if it is even or multiplied by 3 and added to 1 if it is odd.  Yes, puzzlers aren’t the only community to dedicate time to solving capricious whims and fancies.

Taking inspiration from this unsolved mathematical puzzle, Dr. Volker Latussek transmogrified the 3n+1 transformation into 3Y+1W with the objective of reducing it to 1 restricted opening square space.

Collatz by Volker Latussek was Hendrick Haak’s exchange puzzle at the International Puzzle Party (IPP) 41.  It consists of 3 Y pentominoes and 1 W pentomino.  The objective is to place all 4 pentominoes in a 5x5 square.  To help you do this, a 5x5 tray is provided.  To further help you, there is a top on the tray to keep pieces from jumping out while you solve the puzzle.  A small slot in the side provides a convenient way to add the pieces to the tray.  Of course it is slightly offset to keep pieces from simply sliding out.

The puzzle is a nicely made ensemble of 3D printing and Laser Cut Acrylic.  The tray is 3D printed and the 4 pieces and top are laser-cut from Acrylic.  The top has 4 sizeable holes that can be used to manipulate the pieces within the tray.  And it can be slid off so that you can easily store the pieces in the tray so that they don’t fall out.  I came to the conclusion a while ago that all the packing puzzles that I print would have a sliding opening and I’m glad to see others embracing this as well.

So my solving experience went like this: This looks like it might work but I’ll be disappointed if it’s that easy.  Nope, the offset opening doesn’t allow that.  OK – maybe it’s like this, but it would still be too easy.  Nope #2.  Maybe I should think about it for a minute.  How about this – Yup.

It’s not difficult, but you do need to determine how the pieces interact with each other to get them correctly into the tray.  And I wasn’t disappointed!

Puzzle Philosophy - Yin Yang

I like puzzles but they aggravate me.  When I’m working on a puzzle, I can’t wait to find the solution.  When I find the solution, I’m disappointed that it’s over.  I expected all these reoccurring conflicting emotions to surface when I pulled (pushed?) Yin Yang out to work on.

Yin Yang was developed by Volker Latussek and made by Pelikan Puzzles.  The box was made using Cherry and has a Maple (Yang) top to contrast nicely with the 6 Wenge (Yin) pieces.  When all the pieces are packed in the box, you end up with a nice digitized taijitu symbol.

Upon inspection, 4 of the Yin pieces are symmetric and the other 2 are not.  These asymmetric pieces are the key Yin and Yang pieces.  I know that I stated that the Wenge pieces were the Yin to the box’s Yang, but even when you separate them, you still have both in each.  Yeah, it doesn’t make a lot of sense, so let’s just agree to call it a principle and keep building on it.   Now that we have our key Yin Yang Yin pieces, as long as they aren't symmetrically situated within the assembly, these mystical keys can be used to affect a transformation between Yin and Yang assemblies.  You’ll also notice that extra effort went into highlighting that each piece was constructed by combining a 2x3 block with a 1x2 block.  I suppose one of those would be the Yin and the other the Yang requiring the key pieces to be formally referred to as the Yin Yang Yin Yang Yin pieces.

Yin and Yang

Since the Yang box encompasses a 4x4x3 space, the objective is to 1) Find a way to make a 4x4x3 shape with the Yin pieces and 2) discover a method for jamming those Yins into the Yang.  And the answers to all your questions are Yes – Are rotations allowed/required, Are there uncrammable assemblies, If I Yin instead of Yang will the pieces get stuck?

I’m embarrassed to admit it took longer than expected to find a 4x4x3 assembly.  Of course once I found it, it had to be the required assembly and I was never going to let it go.  I could see how the first 3 pieces would come out and only had to figure out how to release the remaining 3.  Surely, taking out (Yang?) / putting in (Yin?) 3 pieces with all that space couldn’t be that hard (With all my experience with 3D apparent packing cubes, it’s a wonder I can still think that).  Needless to say, I spent a considerable amount of time experimenting with some amazing cramming techniques with those obstinate pieces until I remembered the principle of the whole thing and immediately solved it.

Off With Her Head! - Guillotine

AAAGGGHHHHHHHH!!!!!!!!!  Off with her head!  The only puzzle that I left unsolved at RPP that tormented me was Guillotine (aka Harun) designed by Volker Latussek.  Just when I thought that I had left all that stress behind me, a package arrived from the UK.  Allard Walker, somehow sensing that I would be starting to recover from my traumatic experience at RPP, strategically mailed me a copy to arrive just in time to prolong the frustration.  I’m lucky to have such good friends.

It turns out that Guillotine was Allard’s exchange puzzle for the Edward Hordern Puzzle Exchange at IPP39 this year.  For those not familiar with the exchange, each participant brings up to 100 copies of a new puzzle design that has not been released in the wild yet, to exchange with the other participants.  This year there were just over 70 participants, so each participant left with 70+ new puzzles.  In addition, a copy of each puzzle is donated to the Lily Library at Indiana University, which now houses more than 34,000 puzzles, generously donated by IPP's founder, Jerry Slocum.

As I explained in A Decade of Puzzling – RPP 2019, when I got back from RPP, I had an epiphany on how to finally solve this bugger.  Overjoyed to finally be able to validate my hypothesis, I eagerly unpacked Guillotine and began to arrange the pieces.  Unfortunately, epiphanies are cheap and this one wasn't worth much.  It wasn't even close.

Guillotine consists of 12 pieces that have to be packed into a 5x5x5 box.  There are 6 planks that are 4x2x1 and 6 additional 4x2x1 planks that have 2x1x1 pieces added on each end (or you can think of it as a 4x2x2 burr piece with a 2x2x1 notch taken out of the middle).  The box has a sliding lid that covers half the box.  It only covers half to allow the pieces to stick out in the unsolved state that it comes in.

After giving up on my erroneous hypothesis, I called it a night to get some sleep before attacking it again.  I resumed the effort again in the morning, and after a short time finally managed to find a solution. Very clever puzzle!

Peter Wiltshire mentioned that there were 2 solutions at RPP, a hard one and an easy one.  I assumed that after all the struggling that I did, I finally managed to find the easy solution.  After another 5 minutes, I found the second solution.  Having both solutions, I’m now assuming that I found the harder one first and that’s why I found the second solution so fast after the first one.

The copy that I played with was made by Eric Fuller and went by the name Harun, which I assume is the name that Volker Latussek gave the puzzle.  Allard’s exchange puzzles were made by ROMBOL GmbH, and I’m told that they sometimes change the names of puzzles for marketing reasons.  So Harun became Guillotine.  Either way, this one is a tough one to get your head into, or maybe I should say out of.

Thank you Allard!