An Attractive Puzzle – MagneTeam

You either like or don’t like apparent cube packing puzzles and MagneTeam is certainly a polarizing puzzle.  This puzzle doubles down on apparent cube packingness, giving you that much more to love or hate.

MagneTeam was designed by Alexander Magyarics and made by Cubicdissection.  The box is made from Ash with a - shaped opening on one side and a + shaped opening on the other.  The Cubicdissection team thoughtfully beveled the 4 inner corners of the + opening for when you poke your fingers inside.  The 4 pieces to be packed are made from Granadillo and Morado.  Why 2 different types of wood for the pieces you ask?  Because there are 2 objectives.  The first objective is to pack the 3 Granadillo pieces into the box making an apparent cube (as a quick reminder, an apparent cube is a shape that looks like a cube but may not completely be or be completely a cube.  It just fills all the openings of the box and looks like one.  You can think of it as a cube wannabe/impostor/impersonator – you get the idea).  Once you have apparent( cubic)ly solved the first challenge, the second challenge is to add the Morado piece and pack all 4 within the box to make another apparent cube.  Since the first challenge was already an apparent cube, chances are that you can’t just simply add the additional piece for a quick win but I’m pretty sure that you guessed that already.

So what do you need to know to solve this puzzle.  On the negative side, it may be daunting if you’re not used to apparent cube packing puzzles and unaware of how to utilize small openings to solve the puzzle.  On the positive side, this puzzle has a big opening that easily permits pieces to enter the box.  Both challenges are fun and can be solved like most apparent cube packing puzzles.  No rotations are required but each has a nice difficulty level: 9.4.4.3 and 7.6.6.  

The biggest challenge for the 3 piece apparent cube was quite unexpected.  The Granadillo looks nice and the Morado looks just as nice, so much so, I couldn’t really tell them apart.  You say Granadillo, I say Morado, let’s call the whole thing indistinguishable.  Maybe it was a third challenge, maybe it was Eric Fuller’s sense of humor, but I spent some time trying to convince myself that one of the pieces was made from a different type of wood.  Of course I wouldn’t deprive you of figuring out which piece it is by mentioning it here.  Although if you are curious, you can check it on the puzzle will be played site.  For the record, I would have been perfectly fine if all the pieces had been made from the same type of wood with the challenge being to find which 3 pieces that still made an apparent cube.  

Two very nice challenges for the price of one.  Well done Alexander!
 

Going Postal - Letter Box

Your task - to stuff a Letter Box with letters until it is completely full.  Oh! - and there are 2664 assemblies.  And you can’t just shove them in, some have to be rotated while they are in the box.  You may even find yourself stuck and frustrated if you’re not careful.

All the letters that you will be stuffing in Letter Box are copies of the L tromino - 9 of them.  They all have to be inserted through the small slot.  When solved, the 9 L trominos will completely fill the Letter Box.

Letter Box, designed by Pit Khiam Goh, was made by Cubicdissection and released in December 2020.  The box is made from Walnut and each of the L trominos is made from a different exotic wood: Wenge, Canxan, Jatoba, Zebra, Bloodwood, Purple Heart, Katalox, Chakte Viga, and Ash.  It’s especially nice that each of the L tromino pieces is made from a single piece of wood.  In addition to the slot that permits the L trominos to pass, there are 2 additional holes that can be used to view and manipulate the pieces within the box.

There are many ways to tackle a puzzle.  Since this puzzle has 2664 assemblies, I wouldn’t recommend an exhaustive approach to try each one until you find one that works.  Even if you found a working assembly, it may not be obvious since the solution requires rotating pieces within the box.  For this puzzle, a better approach would be to start at the other end and envision how the pieces would be removed from the box.  How would the first piece come out?  Then with the first piece out, how would the second piece come out?  Using this approach, I was able to find a workable assembly and determine the rotations needed to solve Letter Box.  Not overly difficult but a very nice challenge.

Need An SD Fix? CD With The - ResQ

I’ve been enjoying the Sequential Discovery (SD) puzzles that have recently been released by layerbylayerpuzzles and when Cubicdissection (CD) announced that they would be releasing an SD puzzle, it caught my interest.  This puzzle, of course, is ResQ.  It is based on Visitor Q by Frederic Boucher and extended by Eric Fuller at CD with additional challenges to extend the storyline.

The puzzle arrives in a very recognizable orange box.  In this case, the box had a black bag attached to it.  This bag contained a black towel, a description of the ResQ backstory and goals, and a Reset Guide.

The towel was a nice touch.  Many SD puzzles are susceptible to having small pieces fall out and roll out of sight.  Most puzzle gatherings aren’t complete without someone swearing at pieces rolling across the floor and groups of people on their hands and knees looking under the furniture.  Oh, how I miss the look of horror on their faces.  CD decided to preempt that experience and provide an official towel to work over.

Inside the orange box is another box nicely made from Hawaiian Koa.  It has various square, rectangle, and rounded slots cut out of it.  Inside the box are pieces made from Canxan, Catalox, Bloodwood, and Chakte Viga.  So far, nothing unusual.  It looks like a typical packing problem with the pieces already packed in the box.

The first odd thing of note is the face of an alien with a strange sparkling hair bun on the top of the puzzle.  Apparently, this is a visitor from another dimension and the goal is to rescue the visitor as well as the components of his spaceship from the void (aka the inside of the box).  There is obviously some time/space warping occurring inside the 3x3x3 box since you need to extract a spaceship (made from Paduak and Walnut), two antenna assemblies, a silver fuel disk, a gold reactor orb, and a six-orb navigation AI module.  I was tracking it all until the end.  How could you possible get a six-orb navigation AI module in there?  Well, I was quite surprised when it unexpectedly showed up – on the towel – as if by magic – out of the void – Uh Oh!

I tried to acquire the ResQ version with all the vortex pieces made from Ebony, but failed.  Somebody was looking out for me.  After solving the puzzle, I’m glad that I had the mixed wood version.  Having all the pieces made from the same wood would make it more difficult and not in a good way.  Of course, that’s just my opinion and if I had acquired an Ebony copy, I’m sure that I would be crowing about that version.  I’ll be interested in hearing about the solve experience from the Ebony owners.

It looks like there was at least one design change after the box was made.  There is a feature that did not appear to be used and caused me to hunt around for something that I may have missed or lost.  I guess that you could just call it a space fish on a long wavelength.

The SD aspects of ResQ are very well done and obviously required a fair amount of experimentation to ensure that everything worked as expected.  However, the real challenge is manipulating the vortex to get things where you need them.  Once you have that figured out, everything is much easier.

I can’t stress enough, that everything is so well made that it looks innocent when in reality, it packs a lot of puzzling in a small amount of space.  The Mechanical Puzzles discord had several messages from puzzlers with difficulty finding out how to unlock the vortex at the beginning since it was so well hidden.

I was initially reluctant to look at the reset instructions to put everything back together, especially since it has a ** SPOILER ** warning, but I’m glad that I took a peak and looked at it.  Although it does have some spoiler information that you wouldn’t want to see before salvaging everything from the vortex, it basically provides the initial configuration of the vortex.  Of course, you can store the vortex in another configuration, but I’d bet that the initial configuration is probably designed to require the maximum number of moves to solve.  Don’t be afraid to use the Reset Guide.  In fact, the final goal recommends using the diagram to reset ResQ.

A Tisket A Tasket, Puzzles In – Akaki’s Picnic Baskets

Basket puzzles are all the rage now.  As one Mechanical Puzzles Discord (MPD) member put it, These baskets are like the current beanie babies of the puzzling world.  We had Two Burrs in a Basket by Logan Kleinwaks, 4L Basket and 5L Basket designed by Koichi Miura, 3 Mushroom Basket puzzles designed by Andrey Ustjuzhanin, and now an entire series of Picnic Basket puzzles by Akaki Kuumeri.

There were originally 10 puzzles in the Picnic Basket series and Akaki offered them for sale on his AkakiKuumeri Etsy Shop.  At the same time, he also kindly made the model files freely available on Thingiverse (Akaki's Picnic (basket packing puzzle series)).  Since then he has added 3 more designs and at least 1 more is currently under development.

The picnic basket puzzle descriptions below are in the order that I printed and solved them.  With almost 3 hours required to print each, there is plenty of time to solve each before the next one is ready.  I printed them directly from the Thingiverse STL files and the fit is perfect.  If you turn the basket upside-down the pieces won’t drop until you lightly tap the basket.  Each model also has a looser fitting option if you find that the pieces are coming out too tight on your 3D printer.  The difficulty levels indicated are those provided by Akaki on Thingiverse or Etsy.

Egg Basket – Akaki Basket #6, Medium.  

Of course, before printing the first set of pieces, I needed to print the basket.  I have to say that the fancier basket model is very well done and imparts a huge cuteness factor to the series.  Many members of the MPD proudly display pictures of their printed baskets.  Egg Basket consists of 4 pieces that have to be placed in the basket.  I found it to be easy, but it is a good introduction of how the pieces can be manipulated and inserted into the basket.  The 3 rotational moves are staples for the entire series.

Wine Basket – Akaki Basket #2, Difficult.  

Wine Basket seems to be a favorite on the MDP and for good reason.  It’s a bit more of a challenge than Egg and makes great use of the handle to trap pieces.  I particularly liked the piece that consisted to 2 connected 2x2 blocks.  It interacts nicely with the basket and other pieces. This puzzle itself is worth printing out the basket along with the Wine pieces even if for some reason there was something wrong with you and you weren’t interested in the entire series.

Sandwich Basket – Akaki Basket #10, Difficult.  

Sandwich is a nice example of how pieces have to work with each other.  I didn’t find it that difficult, but it was a nice solve.  The reason I started with Egg, Wine, and Sandwich was the Cubicdissection announcement that these 3 puzzles will be made as a set and included in an upcoming release.  Eric Fuller also indicated on the MPD that the basket for the CD release would be compatible with the puzzle pieces from Akaki’s Etsy shop.  A very nice gesture for those who will want to acquire additional pieces to enlarge the set.  Of course, it would be even nicer to acquire the remaining sets of pieces in different exotic woods.  After having done the entire series, I agree that these 3 make a nice set although I may have been tempted to swap out Egg for one of the more difficult ones, but that’s just my preference.

Subway Basket – Akaki Basket #5, Difficult.  

Unfortunately, I chose to do Subway after Sandwich.  Please avoid doing that.  They are similar and use the same solving method.  The only real difference is that Subway combines 2 of the pieces to make it a 3 piece puzzle.  There is certainly nothing wrong with it, but it was a quick solve for me right after Sandwich.

Coffee Basket – Akaki Basket #9, Medium.  

As a medium puzzle, Coffee Basket is pretty straight-forward.  Nothing really difficult and I didn’t find it as interesting as Egg.  I should mention that I only printed one basket and have been swapping the pieces in and out so there is no real need to skip any of these puzzles when it takes less than 3 hours to print each.  Akaki also provides the option on his Etsy site to buy each puzzle with or without the basket.

Hamburger Basket – Akaki Basket #13, Medium.

Luckily, I had a number of filament colors so that I could print each puzzle in a different color.  Unfortunately, I ended up with some odd things like blue hamburgers.  Hamburger Basket is one of the 3 puzzles added after the first set of 10 on Thingiverse.  Personally, with only 3 pieces and some obvious piece positions, I thought this one was easy.   However, I should add that Hamburger Basket is a favorite on the MPD.
 

Fruit Basket – Akaki Basket #4, Easy.  

When I mentioned to my Non-Puzzling Significant Other (NPSO), a full-fledged member of the Spouses of Addicted Puzzlers (SOAP) club, that I was only planning to make the difficult and medium puzzles, she said WAIT A MINUTE!.  Witness the power of the cuteness factor!  So I had to change my plan and not only make them all, but print one of the easy ones next.  Enter Fruit Basket.  Not expecting much, I set about putting the pieces in the basket and didn’t find it as easy as some of the more difficult puzzles.  It also added another movement in the basket solving arsenal.  I decided that this one was a bit underrated or at least no simpler than some of the medium level puzzles and decided not to offer it as a starting point for my innocent NPSO.

Chicken Basket  – Akaki Basket #12, Medium.  

Chicken is not that difficult and expands on the new move from Fruit Basket.  It also has one more trick up its sleeve to make it interesting.  Just when it looks like it would be trivial, the pieces conspire to get in each other’s way and require you to figure out some additional steps to stuff that chicken in the basket.

Chocolate Basket – Akaki Basket #1, Difficult.

Chocolate Basket is a great design.  The only problem that I had with it is that by this time I was familiar with all the moves required and it was a quick solve.

Salmiakki Basket – Akaki Basket #11, Medium.  

Before now, I’d never heard of Salmiakki.  I assumed that it was some sort of Japanese sushi dish but it turns out that it is a licorice made with salmiak salt.  Who knew?    I found Salmiakki Basket to be tricky and it had me going in circles for a bit.  Just when you think there couldn’t be something new.

Cake Basket – Akaki Basket #8, Easy.  

As the name implies, this one was cake and lived up to its rating.  You could easily skip this one.  However, if you are like me and only printed a single basket for all the puzzles, it’s not really a big deal to print out the pieces for a particular puzzle.  On the MPD there are puzzlers in both camps with some printing a basket for each puzzle or like me a single basket for all the puzzles.  There are some that have even printed out a scaled up copy of the basket to hold the pieces for all the puzzles when not in the official basket.

Ice Cream Basket – Akaki Basket #7, Easy.  

Ice Cream is definitely easy and a nice follow-on to cake.  It even introduces a new type of move, which is impressive considering that it was the twelfth one that I’ve done.  The variety of different types of movements is surprising.  Akaki mentioned that his design process consisted of developing a piece with interesting movements/rotations and then subtracting that piece from the cube space to then be further cut up into other interesting pieces.  Of course, given the interaction among the pieces, this description is a very simplistic summary of a more intricate process.

Vegetable Basket – Akaki Basket #3, Easy.  

And then there was Vegetable Basket.  I found this easy puzzle to be the most difficult of the bunch.  I’m embarrassed to say that I spent more than an hour trying to solve this one.  Although the move count is not high, there are multiple cube assemblies that can be made with the four pieces requiring you to find the correct one as well as put it in the basket.  I kept doing the same things over and over and ending up with the same assemblies that I couldn’t convince myself would go in the basket.  It was insane and I was becoming so myself.  I was pondering a new name like Basket Case for this puzzle.  Thanks to my NPSO for not letting me miss out on the frustration.  For an easy puzzle, I highly recommend this one.

In general, I found the easy puzzles to be a mixed bag with 2 easier than the medium puzzles and 2 harder than the medium puzzles.  The 2 harder ones were so under constrained that they were more difficult to solve.  Unlike some of the difficult ones that were so constrained that they were easy to solve.  I’d certainly be interested in hearing about other people’s experiences with these puzzles in the comments.

Akaki has mentioned on the MPD that he is working on an even more difficult “Boss” puzzle (codename: Peppermint Basket) to cap the series.  Of course, the MPD members have indicated that they don’t want to see the series capped, but continually expanded beyond the “Boss” puzzle.  Other well-known puzzle designers on the MPD, like William Hu, are also considering contributing to the series (codename: Nachos Basket).  I’m looking forward to seeing how this series evolves, especially after seeing some photos involving angled half-cubes.

Finally, a big thank you to Akaki for making the Picnic Basket models available to the puzzle community!

It’s a Feature – Inaccurate Burr

What happens after you’ve completed 336 burr puzzle pieces only to realize that you’ve completely flubbed the measurements for the cuts – You call it a feature and charge extra!

Pure marketing genius!  And Eric Fuller is that marketing genius behind Cubicdissection’s branding drive.  Pavlov would be proud to see the puzzling masses salivating over the sight of cardboard boxes secured with orange tape.

When Inaccurate Burr was released in 2016, acquiring a copy was a no-brainer.  What’s not to like?  Designed by Junichi Yananose, make by Eric Fuller at Cubicdissection, and Oh, that beautiful Marblewood!

As per my wont, I quickly disassembled the burr without paying too much attention so that I could attack it as an assembly process.  Only when I had it apart did I realize that I was a tad impetuous and should have looked at it first to see why it was called Inaccurate Burr.  I’d like to be able to say that this is a rare occurrence, but it seems to occur often.  My favorite is pulling out pieces of a puzzle that I had disassembled years ago with no idea what puzzle it is or what shape it’s supposed to make.  And of course, the ultimate experience is when you acquire pieces of a puzzle with unknown origins.  Some puzzlers even have their friends disassemble multiple puzzles and mix the pieces to provide the penultimate solving experience transcending the normal single puzzle experience, which as you already know, is quite powerful.

After my initial, what is it supposed to look like? panic, I started to look at the pieces.  Two pieces immediately stood out.  Instead of having cuts that were half way through the pieces, one had cuts two-thirds deep and the other one-third deep.  The natural assumption would be that these two pieces make one of the three pairs of the 6-piece burr and that this offset led to the name’s genesis.

With the first pair is established, it is a fairly logical progression to add the others.  One piece of the pair has a knob that sticks out one-third and one of the remaining four pieces has a notch one-third deep.  Three pieces down!  Another piece can’t go anywhere except to be the third piece’s mate.  Four pieces down!  For the last pair, you would be forgiven if you tried to put them in backwards.  It looks like either way may work, but if you pick the wrong one, you will hopefully, quickly realize and rectify the incorrect positioning.  Once you get it correct, it ONLY takes 9 moves to insert that last piece and get the puzzle into its final inaccurate shape.

This puzzle may not be difficult but it is nice addition to a 6-piece burr collection and looks awesome in Marblewood.

Mangling Wood With Style - In Brackets

It’s 2021!  A brand new year!  And just like how everyone who wants to forget the prior year, I’d like to leave behind my initial puzzle making attempts.  Then along came Cubicdissection’s announcement that In Brackets by Sam Cornwell (released in 2009) may be rereleased in January 2021.  

Back in 2012, when I saw In Brackets on Cubicdissection’s website, Eric Fuller was mostly releasing new puzzle designs and there was no indication that he would change his approach and rerelease prior puzzles.  I also didn’t see any indication that In Brackets was going to be made by anyone else.  So I did what any other wood mangler would do and headed to the garage.

In addition to the nice photos of In Brackets on Cubicdissection’s website, Eric Fuller included the following details in his description (https://cubicdissection.com/products/in-brackets):

Crafted from fine peruvian walnut and zebrawood, this puzzle has a very nice fit and feel.  The cube is precise but not tight, and the brackets have a good feel with a generous .012 offset from the cube. Brackets are constructed with finger joints at the edges to insure strength and long life.

As a new woodworker, I loosely translated this into my own frame of reference and ended up with the following build process:

Cobbled together with Red Oak and Cherry so it won’t fall apart in your hands.  There’s enough slop in there to ensure that the cube doesn’t get wedged in the brackets.  Butt joints are used on the brackets to expedite the build process.

Armed with a solid plan, I marched into the garage and mangled some wood into something that could be reminiscent of In Brackets if you squint at it just right.  This puzzle was one of the first victims of my newly purchased miter saw many years ago and is now part of my eyesore puzzle collection.  However, it is a fully functional puzzle and served to satisfy that particular puzzle itch.  

Before, I describe the puzzle, please keep in mind that these comments are based on my homemade copy and a professionally made version may give you a whole other, potentially sublime, experience.

The puzzle consists of 3 pieces that make a 3x3x3 cube with 3 voids and 3 brackets that hold the cube together.  As a puzzle, this one is pretty easy.  Taking it apart is trivial, but to be fair, if the puzzle were tight or incorporated a magnet, it would provide more of a challenge to find the first move.  Of course, since I had made the pieces, it started out as an assembly challenge.

Assembling In Brackets is not difficult and would make a good challenge for new puzzler.  It’s easy to create a cube from the 3 internal pieces.  Once you have the cube, it doesn’t take a lot of analysis to determine how to add the brackets.  Since any 2 brackets can be trivially added to the cube, you only have to determine where those 2 brackets have to go to allow the cube pieces to move into a configuration that permits the insertion of the final bracket.

If you are looking for a challenging puzzle and have been enjoying the wave of Turning Interlocking Cubes (TICs), this one is not for you.  However, if you want an attractive, very approachable puzzle with a novel approach, you will enjoy In Brackets.  Cubicdissection’s 2009 version of In Brackets looks fantastic and I’m sure that if it’s rereleased, it will look even better.

A Christmas Present For You 2! – Penultimate Burr Box Set 7-Piece Burr Challenges

I know that by now, most of you have completed all 898 Penultimate Burr Box Set challenges including last year’s Christmas Challenge and were wondering what to do with that old dusty box.  In the event that you haven’t sold, traded, or regifted your stick collection, there is now an entirely new set of challenges for you.

6-Piece burrs have been all the rage with everyone amassing large quantities of this venerable puzzle and all its variants.  However, the 6-piece burr is so 2020!  As we leave 2020 behind us, a new set of challenges is needed for 2021.  Enter the 7-piece burr, 6’s lesser known brother.

Of course, there are multiple ways to construct a shape with 7 burr pieces.  With the construction that I chose, BurrTools indicated there are 3,232,523 assemblies and 184,687 solutions.  In English, this means that there are 3,232,523 ways that 7 of the burr pieces from the set can make the 7-piece burr shape, but only 184,687 of these assemblies can be taken apart.  Further analysis indicates that 471 of the solutions are unique.  A unique solution is one where the 7 pieces cannot be rearranged to make the 7-piece burr another way.  Of the 471 unique solutions, 38 are level 2 and the remaining 433 are level 1, where the level is the number of moves required to remove the first piece or set of pieces from the puzzle.

I haven’t completed all 471 solutions yet.  OK, I’ve only done the first one so far.  I didn’t find it that difficult but it does require some different thinking.  In the 6-piece burr, all the pieces basically perform the same function, however in the 7-piece burr, 2 of the pieces now function differently.  This can be leveraged during the solution process.

First Attempt: During my first attempt, I managed to assemble 6 of the pieces into an assembly that would have accommodated the final piece, but there was no way to add that piece.  Some extra thinking was required to find an assembly that can be constructed.  

Second Attempt: My second attempt was close, but the parity of the last piece was wrong for the assembly that I was constructing.  (sheesh, what does he mean by parity? – Basically, the assembly had a hole on the left and the piece had a cube on the right).  

Final Attempt: Assuming that the final piece was correct, I rebuilt the assembly with a parity to match the piece and it finally went together.

Some of the puzzles will be very similar to each other and you may want to cherry pick ones that look interesting.  Then again, 2 puzzles may only differ by 1 piece yet be entirely different. If you would like to give these a try yourself, the pieces required for the unique solutions are below.  Hopefully, these challenges will keep your set from joining the yuletide log.

Level 2 Unique Solution Piece Sets:

6,7,9,10,15,17,19	6,7,9,10,15,17,20	5,6,7,12,17,22,24	5,6,7,10,21,22,24
5,6,7,12,15,20,22	5,6,7,15,17,23,24	3,5,6,7,8,22,24	6,7,9,10,11,17,24
5,6,7,9,16,19,24	5,7,10,12,15,18,24	5,6,7,9,19,24,25	5,6,7,10,15,23,24
5,7,10,12,17,22,24	5,6,7,9,17,24,26	4,5,6,7,8,22,24	4,5,6,7,11,15,24
5,6,7,15,17,18,24	5,6,7,12,15,17,22	5,6,7,15,17,20,22	6,7,9,10,14,15,19
6,7,9,10,14,15,20	5,6,7,10,20,22,24	5,6,7,10,19,22,24	5,6,7,12,15,23,24
5,7,10,15,17,18,24	6,7,10,11,12,15,17	5,7,10,12,15,17,22	5,7,10,12,15,17,24
5,6,7,10,15,22,26	5,6,7,10,15,24,26	6,7,10,12,14,15,17	5,6,7,17,20,22,24
5,6,7,9,17,23,24	3,5,6,7,11,15,24	5,6,7,12,20,22,24	4,5,6,7,10,15,24
5,6,7,12,15,18,24	6,7,10,13,14,15,17

Level 1 Unique Solution Piece Sets:

5,7,9,10,18,21,24	5,7,9,10,15,18,23	5,7,9,10,15,18,26	5,7,10,11,15,16,19
5,7,10,11,15,16,22	5,7,10,11,15,16,24	1,5,7,8,10,18,22	4,6,7,8,9,10,22
4,6,7,8,9,10,24	1,3,5,6,9,10,15	5,6,7,15,17,21,23	5,6,7,15,17,21,26
5,7,9,10,13,16,19	4,5,7,9,10,15,22	4,5,7,9,10,15,24	5,7,9,10,13,16,20
6,7,9,10,13,15,23	6,7,9,10,13,15,26	1,5,7,9,10,12,19	1,5,7,9,10,12,20
5,6,10,13,17,24,25	5,6,7,11,13,24,26	5,7,10,11,15,18,19	1,5,6,9,10,21,24
5,7,10,14,15,17,18	5,7,9,10,14,19,22	5,7,9,10,14,19,24	5,7,9,10,14,19,25
5,7,10,14,15,17,21	5,6,10,14,15,16,17	1,5,6,10,15,16,17	1,4,5,6,9,10,15
5,7,9,10,16,21,24	5,7,9,10,13,18,19	5,7,9,10,13,18,20	1,5,6,10,14,15,18
1,5,6,10,14,15,19	1,5,7,9,10,14,19	5,7,10,11,13,16,24	1,5,7,9,10,14,20
5,6,10,11,15,19,22	5,6,10,11,15,19,24	5,7,10,11,12,15,21	5,7,10,14,15,19,22
5,7,10,14,15,19,24	5,7,10,14,15,19,25	5,7,9,10,11,16,19	5,7,9,10,11,16,20
5,6,10,14,15,18,22	5,6,10,14,15,18,24	5,7,10,11,17,22,24	3,5,6,7,8,24,25
1,5,7,10,15,16,17	6,7,9,10,11,15,23	6,7,9,10,11,15,26	5,6,7,13,15,18,23
5,6,7,13,15,18,26	5,6,9,10,21,22,24	5,7,10,11,13,18,24	5,7,9,10,15,22,23
5,7,9,10,15,22,26	4,5,6,7,9,17,19	1,5,7,10,14,15,18	1,5,7,10,14,15,19
1,5,7,9,10,16,22	1,5,6,10,11,12,22	4,5,6,7,9,17,20	5,7,9,10,12,19,24
5,6,9,10,17,19,22	5,6,9,10,17,19,24	5,6,9,10,17,19,25	5,7,9,10,14,21,24
1,5,7,8,10,22,25	5,7,9,10,11,18,19	5,7,9,10,11,18,20	4,5,6,7,8,16,24
6,7,8,9,10,22,23	6,7,8,9,10,22,26	5,6,9,10,16,18,24	5,6,7,9,21,23,24
1,5,6,10,12,15,16	1,5,6,10,12,15,19	1,5,6,10,12,15,21	5,7,9,10,13,20,25
5,6,9,10,18,20,24	4,6,7,8,10,14,15	5,7,10,12,15,16,21	5,7,10,12,15,16,22
5,7,10,12,15,16,24	5,6,7,9,20,22,23	5,6,7,9,20,22,26	1,5,6,7,9,13,23
5,6,7,13,17,22,23	1,5,6,7,9,13,26	5,6,7,13,17,22,26	5,7,9,10,15,24,26
1,5,7,9,10,18,24	1,5,6,10,11,14,24	1,5,7,10,11,12,22	5,7,10,11,15,22,25
5,7,10,12,14,15,21	5,7,10,15,17,19,22	5,7,10,15,17,19,24	5,6,10,15,17,18,19
1,5,6,8,10,16,24	1,5,7,8,10,24,25	6,7,8,9,10,24,26	1,4,5,6,7,9,12
5,6,10,15,16,17,18	5,6,10,15,16,17,19	1,5,7,10,12,15,16	1,5,7,10,12,15,19
5,6,10,15,16,17,22	5,6,10,15,16,17,24	5,6,10,15,16,17,25	5,6,9,10,18,22,24
1,5,7,10,12,15,21	5,7,10,12,15,18,21	3,5,6,7,9,12,19	3,6,7,8,10,15,17
3,5,6,7,9,12,20	1,5,7,10,11,14,24	5,7,10,11,15,24,25	4,5,6,7,15,17,19
1,5,6,8,10,18,22	1,5,6,7,8,14,23	1,5,6,7,8,14,26	5,6,10,12,14,16,24
3,5,6,7,15,16,17	5,7,9,10,11,20,25	5,6,9,10,16,20,24	4,5,6,9,10,15,16
4,5,6,9,10,15,17	4,5,6,9,10,15,18	4,5,6,9,10,15,22	4,5,6,9,10,15,24
4,5,6,9,10,15,25	3,5,6,9,10,14,15	5,6,10,11,14,22,24	5,6,9,10,18,24,25
5,6,9,10,12,16,19	1,5,6,9,10,12,19	5,6,10,12,15,19,22	5,6,10,12,15,19,24
5,7,10,11,13,22,24	5,6,7,9,14,19,23	5,6,9,10,12,16,20	5,6,10,12,15,19,25
5,6,7,9,14,19,26	1,5,6,9,10,12,20	1,5,7,9,10,20,22	1,5,7,9,10,20,24
5,6,7,11,15,19,23	5,6,7,11,15,19,26	5,7,10,15,17,21,22	5,7,10,15,17,21,24
5,7,10,15,17,21,25	5,6,7,14,15,18,23	5,6,7,14,15,18,26	5,6,10,12,14,18,24
4,5,6,7,12,14,24	5,6,9,10,16,22,24	4,6,7,9,10,12,15	5,6,9,10,13,19,22
3,6,7,9,10,11,15	5,7,10,13,17,18,22	5,6,9,10,12,18,19	1,5,6,9,10,14,19
5,6,9,10,12,18,20	5,7,10,11,13,24,25	1,5,6,9,10,14,20	4,5,6,7,13,17,24
5,7,10,13,16,17,22	5,7,10,13,15,16,18	5,7,10,13,15,16,19	3,5,6,7,12,15,16
3,5,6,7,12,15,19	3,6,7,9,10,13,15	5,6,9,10,15,23,24	3,5,6,7,9,18,24
1,5,6,9,10,16,22	5,6,10,12,15,21,22	5,6,10,12,15,21,24	5,6,9,10,11,19,22
5,6,9,10,11,19,24	1,5,6,8,10,22,25	5,6,7,9,13,20,23	5,6,7,9,13,20,26
4,5,6,7,8,24,25	5,7,10,13,15,18,19	5,7,10,13,15,18,22	5,7,10,13,15,18,24
5,7,10,13,15,18,25	5,7,9,10,21,22,24	4,5,6,7,11,17,24	1,5,6,9,10,18,24
5,6,9,10,12,20,25	5,7,9,10,17,19,22	5,7,9,10,17,19,24	6,7,8,10,14,15,23
6,7,8,10,14,15,26	5,6,7,11,15,23,25	3,5,6,7,10,15,24	1,5,6,8,10,24,25
5,6,7,8,22,23,25	5,7,9,10,16,18,22	1,3,5,6,7,9,12	5,6,10,13,15,19,22
5,6,10,13,15,19,24	5,7,9,10,18,20,22	5,7,9,10,21,24,25	5,7,9,10,15,17,23
5,7,9,10,15,17,26	5,7,10,13,17,22,25	3,5,6,7,9,20,24	5,7,9,10,14,16,19
5,6,7,9,11,20,23	5,6,7,9,11,20,26	5,7,9,10,14,16,20	5,6,7,11,15,25,26
5,6,7,8,22,25,26	5,6,10,12,14,24,25	5,6,10,11,17,18,24	5,6,7,11,14,24,26
5,7,9,10,18,22,24	5,6,10,11,16,17,24	1,5,7,9,10,11,19	4,5,6,7,9,12,19
1,5,7,9,10,11,20	4,5,6,7,9,12,20	5,6,7,11,13,23,24	1,5,6,9,10,20,22
1,5,6,9,10,20,24	5,7,10,11,15,17,21	5,7,9,10,14,18,19	5,7,10,14,15,16,18
5,7,10,14,15,16,19	5,7,9,10,14,18,20	5,6,10,11,15,16,22	5,6,10,11,15,16,24
5,6,7,11,12,22,23	5,6,7,11,12,22,26	5,6,7,10,15,16,20	5,7,10,11,14,16,24
5,7,9,10,16,20,22	3,5,7,9,10,15,22	3,5,7,9,10,15,24	5,6,7,8,18,22,23
5,6,7,8,18,22,26	1,5,7,9,10,13,19	1,5,7,9,10,13,20	5,6,7,12,15,21,23
5,6,7,12,15,21,26	5,7,10,11,15,19,22	5,7,10,11,15,19,24	5,7,10,11,15,19,25
6,7,9,10,12,15,23	6,7,9,10,12,15,26	5,7,10,14,15,18,19	5,7,10,14,15,18,22
5,7,10,14,15,18,24	5,7,10,14,15,18,25	1,5,6,10,15,17,19	1,5,6,10,15,17,21
5,6,7,10,15,18,20	5,7,10,11,14,18,24	5,7,9,10,16,22,24	5,7,9,10,16,22,25
5,7,9,10,13,19,22	5,7,9,10,13,19,25	1,5,7,9,10,15,23	1,5,7,9,10,15,26
5,6,9,10,17,18,19	5,6,9,10,17,18,20	1,5,6,10,13,15,18	1,5,6,10,13,15,19
5,7,10,11,12,16,22	5,7,9,10,14,20,25	5,6,7,12,14,22,23	5,6,7,12,14,22,26
5,6,9,10,16,17,19	5,6,10,14,15,19,22	5,6,10,14,15,19,24	5,6,9,10,16,17,20
5,6,10,11,17,22,24	1,5,7,10,15,17,19	1,5,6,10,12,14,22	1,5,6,10,12,14,24
5,6,7,13,15,19,23	5,6,7,13,15,19,26	1,5,7,10,15,17,21	1,3,5,7,9,10,15
1,5,7,9,10,17,19	5,7,9,10,15,23,24	5,7,9,10,15,23,25	1,5,7,9,10,17,20
1,5,6,10,11,13,24	4,5,6,7,9,18,24	1,5,6,10,13,17,22	5,7,10,15,17,18,21
1,5,6,10,13,17,24	5,7,10,11,12,18,22	1,5,6,7,8,11,23	1,5,6,7,8,11,26
1,5,7,10,13,15,18	1,5,7,10,13,15,19	5,7,9,10,11,19,22	5,7,9,10,11,19,24
5,7,9,10,11,19,25	6,7,8,9,10,23,24	1,4,5,6,7,8,17	5,6,7,9,21,24,26
5,6,10,11,17,24,25	5,7,10,15,16,17,21	5,7,10,15,16,17,22	5,7,10,15,16,17,24
5,7,9,10,13,21,24	1,5,7,10,12,14,22	1,5,7,10,12,14,24	4,6,7,8,10,15,17
5,6,10,12,15,16,18	5,6,10,12,15,16,19	1,5,6,10,11,15,16	1,5,6,10,11,15,19
3,6,7,8,10,14,15	5,7,9,10,15,25,26	5,6,10,12,15,16,22	5,6,10,12,15,16,24
5,6,10,12,15,16,25	1,4,5,7,9,10,15	1,5,6,10,11,15,25	5,6,9,10,17,20,25
5,6,9,10,20,24,25	1,5,7,10,11,13,24	5,6,7,11,15,16,23	5,6,7,11,15,16,26
5,7,10,12,14,16,22	4,5,6,7,15,16,17	3,6,7,8,9,10,22	3,6,7,8,9,10,24
5,6,10,11,15,22,25	5,6,10,15,17,19,22	5,6,10,15,17,19,24	5,6,10,15,17,19,25
1,5,7,10,13,17,22	1,5,7,10,13,17,24	5,6,7,10,15,22,23	4,5,6,9,10,14,15
5,7,10,11,14,22,24	5,7,10,12,13,15,21	5,7,10,12,15,19,22	5,7,10,12,15,19,24
5,6,10,12,15,18,19	1,5,6,9,10,11,19	1,5,6,9,10,11,20	1,5,6,10,11,17,24
4,5,6,7,9,20,24	1,5,7,10,11,15,16	1,5,7,10,11,15,19	1,5,7,10,11,15,25
5,6,9,10,14,19,22	5,6,9,10,14,19,24	5,7,10,12,14,18,22	5,6,10,11,15,24,25
3,5,6,7,15,17,19	5,7,9,10,11,21,24	4,6,7,9,10,11,15	5,7,10,11,14,24,25
3,5,6,9,10,15,16	3,5,6,9,10,15,17	3,5,6,9,10,15,18	3,5,6,9,10,15,22
3,5,6,9,10,15,24	3,5,6,9,10,15,25	1,5,6,9,10,13,19	1,5,6,9,10,13,20
1,5,7,9,10,21,24	5,6,10,11,13,22,24	1,5,7,10,11,17,24	5,6,7,9,16,22,23
5,6,7,9,16,22,26	5,6,7,14,15,19,23	5,6,7,14,15,19,26	5,6,7,9,13,19,23
5,7,10,11,12,22,25	5,6,7,9,13,19,26	4,5,6,7,12,15,16	4,5,6,7,12,15,19
5,6,10,15,17,21,22	5,6,10,15,17,21,24	4,6,7,9,10,13,15	3,5,6,7,12,14,24
3,6,7,9,10,12,15	5,6,9,10,15,22,23	5,6,9,10,15,22,26	5,7,10,12,15,21,22
3,5,6,7,9,17,19	5,6,10,13,17,18,24	5,7,10,12,15,21,24	5,7,10,12,15,21,25
3,5,6,7,9,17,20	5,6,9,10,12,19,24	5,6,9,10,12,19,25	1,5,6,9,10,15,23
1,5,6,9,10,15,26	5,6,7,9,14,20,23	5,6,7,9,14,20,26	3,5,6,7,13,17,24
5,6,10,13,16,17,24	3,5,6,7,8,16,24	5,7,10,13,15,17,21	5,6,9,10,15,24,26
1,5,6,9,10,17,19	1,5,6,9,10,17,20	5,6,7,9,11,19,23	5,6,7,9,11,19,26
5,7,10,12,14,22,25	1,3,5,6,7,8,17	5,7,10,13,15,19,22	5,7,10,13,15,19,24
5,7,10,13,15,19,25	5,6,10,13,15,18,22	5,6,10,13,15,18,24	5,7,9,10,15,16,23
5,7,9,10,15,16,26	6,7,8,10,15,17,23	6,7,8,10,15,17,26	3,5,6,7,11,17,24
5,7,9,10,20,22,25	5,7,9,10,14,15,23	5,7,9,10,14,15,26	1,5,7,8,10,16,24
5,6,7,11,14,23,24