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SOMA News |
22 Sep 2016
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This interesting enigma was first published in the second issue of The Soma Addict. There is said to be a way to form the cube in which it can be stood atop a "pedestal" no wider than the bottom center cube. Can you find this solution?!?
Now, there are actually three ways of understanding this challenge.
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This interesting information about the Soma Cube was published in Martin Gardner's Mathematical Games column in July, 1969.
Martin Gardner notes in the article that there are 240 different ways the cube can be formed not counting rotations and reflections as being different. This figure was established in 1962 by John Conway and M.J.T. Guy, who were mathematicians at the University of Cambridge. In analyzing his work, Conway discovered the curious fact that only one of the 240 solutions allows the Soma cube to be balanced on a pedistal that touches only the central square of the cube's 3 by 3 unit base. This solution is diagrammed below. The central square of the bottom layer is the one that rests on the pedistal.
TOP MIDDLE BOTTOM 677 557 524 667 364 524 311 314 322
Are there more solutions to this problem? Mr. Collins conjectures that any of the flat 4-cube pieces ( 2,3, and 4 ) might serve as the balance piece. It only remains to be shown that the 3 piece can serve this function.
TOP MIDDLE BOTTOM 221 266 246 711 553 446 773 753 453
551 466 226 751 451 426 773 733 423and
166 446 244 117 356 222 377 357 355...I tried it both ways. It really works!
/122/552/542 /116/376/544 /366/377/374 /322/552/542 /336/176/544 /366/177/174
Standardization of solutions by bottom piece: Notice that when reflecting one solution onto another, pieces 5 & 6 must be swapped. Solution #1: #2: #3: #4: #5: #6: #7: #8: #9: #10: #11: #12: #13: 322 322 773 773 663 663 113 553 551 333 333 774 113 Top 311 337 711 733 633 611 517 753 751 735 711 744 766 341 377 221 223 223 221 577 773 773 775 771 143 776 552 552 763 761 771 773 433 466 466 466 436 766 413 Midt 376 566 663 661 671 673 466 453 451 455 466 553 463 446 117 255 255 255 255 557 711 733 711 755 113 755 572 452 463 461 471 473 223 226 226 226 226 226 223 Bottom 577 446 445 445 445 445 426 426 426 426 425 526 425 466 146 245 245 245 245 426 421 423 421 425 523 425 Bottom .7. 4.. 4.. 4.. 4.. 4.. 22. 22. 22. 22. 22. 22. 22. piece .77 44. 44. 44. 44. 44. .2. .2. .2. .2. .2. .2. .2. ... .4. .4. .4. .4. .4. .2. .2. .2. .2. .2. .2. .2. Reflections #1: #2: #3: #4: #5: #6: #7: #8: #9: #10: #11: #12: #13: 112 223 377 377 355 355 311 366 166 333 333 477 311 Top 412 733 117 337 335 115 716 367 167 637 117 447 557 333 773 122 322 322 122 776 377 377 677 177 341 577 552 266 357 157 177 377 334 554 554 554 534 557 314 Midt 476 556 355 155 175 375 554 364 164 664 554 366 354 436 711 662 662 662 662 766 117 337 117 667 311 667 572 264 354 154 174 374 322 522 522 522 522 522 322 Bottom 577 544 644 644 644 644 524 524 524 524 624 526 624 466 541 642 642 642 642 524 124 324 124 624 326 624 Bottom .7. ..4 ..4 ..4 ..4 ..4 .22 .22 .22 .22 .22 .22 .22 piece .77 .44 .44 .44 .44 .44 .2. .2. .2. .2. .2. .2. .2. ... .4. .4. .4. .4. .4. .2. .2. .2. .2. .2. .2. .2.Interestingly enough, John Conway's solution which he thought was unique was the ONLY solution which I had NOT already found!
Solution #1: #2: 774 477 Top 712 117 112 122 744 447 Middle 655 655 662 662 345 345 Bottom 335 335 362 362 Reflections (pieces 5 and 6 must be swapped) #1r #2r 112 122 Top 712 117 774 477 552 552 Middle 566 566 744 447 352 352 Bottom 336 336 346 346Solution #1 is (No.33 in N990201) and Solution #2r is (No.53 in N990201).
2 2 3 5 3 3 5 5 3 2 7 7 5 7 4 6 6 4 2 7 4 6 1 4 6 1 1Piece 2
1 1 3 5 5 3 5 5 1 3 3 3 3 3 3 7 7 4 1 1 3 3 3 3 6 6 3 6 6 3 3 3 3 1 1 3 1 1 6 5 1 7 7 5 3 7 5 1 7 3 5 7 1 1 7 4 4 7 6 6 5 3 7 7 3 3 7 1 1 7 6 6 7 3 3 7 6 6 5 7 7 7 7 3 7 7 3 7 7 5 7 7 1 1 4 3 7 7 6 5 7 7 7 7 3 7 7 1 7 7 6 7 7 3 7 7 3 4 3 3 4 6 6 4 6 6 4 6 6 4 3 6 7 6 6 4 1 3 4 1 1 4 6 1 4 6 3 4 3 1 4 1 6 4 1 6 4 6 6 4 5 3 4 5 1 4 5 5 4 6 6 5 5 3 4 6 3 4 6 6 4 6 1 4 6 3 4 6 1 4 6 6 4 5 5 5 5 7 7 1 1 7 3 3 7 1 1 7 5 5 1 1 3 7 5 5 5 5 7 7 5 5 7 5 5 7 5 5 7 5 5 7 3 3 2 2 3 2 2 6 2 2 6 2 2 6 2 2 6 2 2 6 2 2 3 2 2 1 2 2 1 2 2 3 2 2 1 2 2 6 2 2 5 4 2 6 4 2 6 4 2 6 4 2 6 4 2 5 5 2 6 4 2 5 4 2 6 4 2 5 4 2 5 4 2 5 4 2 5 4 2 5 4 2 6 4 2 1 4 2 3 4 2 1 4 2 5 5 2 3 4 2 5 4 2 6 4 2 5 4 2 5 4 2 5 4 2 5 4 2 3Piece 3
7 7 4 4 7 7 7 1 2 1 1 7 1 1 2 1 2 2 7 4 4 4 4 7 6 5 5 6 5 5 6 6 2 6 6 2 3 4 5 3 4 5 3 3 5 3 3 5 3 6 2 3 6 2Piece 4
3 2 2 6 6 3 6 6 3 1 2 2 7 7 3 7 7 3 3 2 2 1 2 2 3 3 7 6 3 3 6 1 1 1 1 7 7 3 3 7 1 1 3 3 6 1 1 6 3 7 7 2 2 3 2 2 1 3 7 7 2 2 3 2 2 1 3 6 6 3 6 6 5 5 2 5 5 1 5 5 3 5 5 2 7 1 1 7 3 3 5 5 2 5 5 2 1 6 6 6 7 1 6 7 3 3 6 6 6 6 5 6 6 5 1 7 6 3 7 6 1 6 7 2 7 7 2 7 7 3 6 7 2 5 5 2 5 5 1 7 7 3 7 7 5 4 2 5 4 1 5 4 3 5 4 2 6 4 1 6 4 3 5 4 2 5 4 2 5 4 4 5 4 4 5 4 4 5 4 4 6 4 4 6 4 4 5 4 4 5 4 4 1 6 4 2 7 4 2 7 4 3 6 4 2 5 4 2 5 4 1 7 4 3 7 4Piece 5 (a mirror of piece 6)
1 7 7 1 1 7 2 2 3 4 4 7 6 6 3 2 5 3 6 4 4 6 5 5 2 5 3Piece 6 (a mirror of piece 5)
7 7 1 7 1 1 3 2 2 7 4 4 3 5 5 3 6 2 4 4 5 6 6 5 3 6 2and Piece 7
3 2 2 3 1 1 3 4 1 5 5 2 3 7 6 4 4 6 5 7 2 5 7 7 4 6 6
Now we know that it is indeed possible to find a SOMA solution, that
will balance on its bottom center cube, so here's a video.
He start by showing a normal cube solution, then destroy it, and show us that a cube
can be built and placed on a pedestal.
The video is from "Hong Kong Middle School - Infinite creativity, toys exhibition."
(I dont understand the language, so I dont really know what he is saying though.)
To see this movie, press the ▶ in the left side of this picture.
The video is also available on YouTube: -
http://www.youtube.com/watch?v=AVCaeBLtB-U "IQ puzzle soma cube (stand up)".
The solutions he show are first a random cube:
222 614 664 214 614 557 333 537 577 Top mid bottomThe balancing cube placed on a pedestal:
113 613 663 557 653 222 577 447 244 Top mid bottom
Back to the Balancing SOMA on a pedestal chapter.
Now one thing is to balance on an externally supplied pedestal,
but what if the balancing object is one of the cubes forming one of the pieces.?
/SOMA166 ;sscube /553/533/113/... /2.4/564/177/.7. /266/264/274/...So, I challenged myself to find a way to balance the blocks with one center point down and one empty space in the otherwise 3x3x3 cube.
/SOMA166 ;sscube_balancing /.41/771/735/... /441/755/635/.3. /422/662/632/... |
Balancing |
Hi Matt: This is a very fine solution. The question of course is "What is a balancing SOMA".
Your idea of following up upon the #166, balancing on one center cube,
and - of course - leaving a hole somewhere... IS great.
A1. A2. A3. A4. A5. 7 7 # 7 7 # 2 4 # 4 4 # 7 4 # Top 2 7 4 4 7 2 2 4 4 2 4 4 7 7 2 2 1 1 1 1 2 1 1 4 2 1 1 1 1 2 6 7 4 4 7 5 2 6 6 6 7 7 4 4 5 Midt 6 5 4 4 6 5 5 5 7 6 5 7 7 6 5 2 5 1 1 6 2 1 7 7 2 5 1 1 6 2 6 6 4 4 5 5 2 3 6 6 6 7 4 5 5 Lower 3 3 3 3 3 3 5 3 6 3 3 3 3 3 3 2 5 5 6 6 2 5 3 7 2 5 5 6 6 2 - - - - - - - - - - - - - - - Balance - 3 - - 3 - - 3 - - 3 - - 3 - - - - - - - - - - - - - - - -
[ 3A | 3B | 3C ] [ 3D ]The whole construction is resting on the pedestal made of cube 3D, and in the first Example 1:
B1. B2. B3. B4. B5. B6. B7. B8. B9. B10. 4 2 2 1 4 7 4 2 2 1 7 7 2 2 4 2 4 # 1 4 7 2 2 7 1 1 4 1 7 7 Top 4 4 1 1 7 7 4 7 1 1 7 4 1 4 4 2 4 4 1 7 7 1 7 7 2 4 4 1 7 4 7 4 1 2 2 2 7 7 1 2 2 2 1 4 # 1 1 4 2 2 # 1 4 # 2 4 # 2 2 # 5 5 2 1 4 4 5 5 2 1 7 4 2 6 6 2 6 6 1 4 4 2 6 6 1 7 7 1 7 4 Midt 7 6 6 6 6 7 4 6 6 6 6 4 5 5 7 5 5 7 6 6 7 5 5 7 6 6 7 6 6 4 7 7 1 2 5 5 4 7 1 2 5 5 1 7 7 1 7 7 2 5 5 1 4 4 2 5 5 2 5 5 5 3 2 6 3 4 5 3 2 6 3 4 2 3 6 2 3 6 6 3 4 2 3 6 6 3 7 6 3 4 Lower 5 3 6 6 3 5 5 3 6 6 3 5 5 3 6 5 3 6 6 3 5 5 3 6 6 3 5 6 3 5 # 3 6 # 3 5 # 3 6 # 3 5 5 3 7 5 3 7 2 3 5 5 3 4 2 3 5 2 3 5 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Balance - 3 - - 3 - - 3 - - 3 - - 3 - - 3 - - 3 - - 3 - - 3 - - 3 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -As mentioned, the crossbeam of piece 3 "T" is called 3A, 3B and 3C.
- 4 - - - - - - - - 4 4 - - 7 - 7 7 - 3 4 - 3 # - 3 7 . . . . 3 . . . .This only allow us the combinations of 4-5-6 and 5-6-7.
7 3 5 6 3 5 6 3 2 C1. C2. C3. C4. C5. 4 1 1 # 4 1 7 4 1 # 4 1 7 7 1 Top 4 4 2 4 4 1 7 7 1 7 7 1 4 7 1 # 4 2 4 2 2 # 2 2 7 2 2 # 2 2 7 7 1 7 7 1 4 4 1 4 4 1 4 7 1 Midt 7 5 5 7 5 5 7 5 5 7 5 5 4 5 5 6 6 2 6 6 2 6 6 2 6 6 2 6 6 2 7 3 5 7 3 5 4 3 5 4 3 5 4 3 5 Lower 6 3 5 6 3 5 6 3 5 6 3 5 6 3 5 6 3 2 6 3 2 6 3 2 6 3 2 6 3 2 - - - - - - - - - - - - - - - Balance - 3 - - 3 - - 3 - - 3 - - 3 - - - - - - - - - - - - - - - -Additional notes
+ 3 5 6 3 5 6 3 + D1. D2. D3. D4. 7 4 1 7 7 1 7 4 1 7 7 1 Top 4 4 1 4 7 1 7 7 1 4 7 1 4 2 2 2 2 2 2 2 2 4 2 2 7 7 1 4 7 1 4 4 1 4 7 1 Midt 7 5 5 4 5 5 7 5 5 4 5 5 6 6 2 6 6 2 6 6 2 6 6 2 # 3 5 4 3 5 4 3 5 # 3 5 Lower 6 3 5 6 3 5 6 3 5 6 3 5 6 3 2 6 3 # 6 3 # 6 3 2 - - - - - - - - - - - - Balance - 3 - - 3 - - 3 - - 3 - - - - - - - - - - - - -
E1. E2. E3. E4. E5. 3 4 # 3 4 # 3 3 3 1 4 3 1 1 3 Top 3 1 1 3 1 2 4 3 # 4 4 # 4 1 # 3 2 2 3 1 2 4 2 2 4 2 2 4 2 2 4 4 5 4 4 5 4 5 1 1 5 3 4 5 3 Midt 3 1 5 3 1 5 4 5 1 1 5 3 4 5 3 6 6 2 6 6 2 6 6 2 6 6 2 6 6 2 4 5 5 4 5 5 5 5 1 5 5 3 5 5 3 Lower 6 7 7 6 7 7 6 7 7 6 7 7 6 7 7 6 7 2 6 7 2 6 7 2 6 7 2 6 7 2 - - - - - - - - - - - - - - - Balance - 7 - - 7 - - 7 - - 7 - - 7 - - - - - - - - - - - - - - - -In some way I feel that these solutions using piece 7 is easier to make, compared to piece 3.
1. 2. 3. 4. 5. 2 2 2 2 2 3 2 2 2 2 1 1 2 2 2 Top 2 # 7 2 # 7 2 # 7 2 # 1 1 # 2 3 7 7 2 7 7 3 7 7 2 2 3 1 1 3 4 4 1 4 5 3 5 5 4 4 4 7 4 4 7 Midt 3 5 1 4 5 3 3 1 4 5 7 7 5 7 7 3 5 7 1 1 7 3 1 7 5 3 3 5 3 3 6 4 4 5 5 3 5 6 6 6 4 4 6 4 4 Lower 6 6 1 4 6 6 5 6 4 6 6 7 6 6 7 3 5 5 4 1 6 3 1 4 5 5 3 5 5 3 - - - - - - - - - - - - - - - Balance - 6 - - 6 - - 6 - - 6 - - 6 - - - - - - - - - - - - - - - -
E6. E6./mirror E7. E7./mirror 3 3 3 3 3 3 1 1 2 2 1 1 Top 4 1 2 2 1 4 4 1 2 2 1 4 1 1 2 2 1 1 3 3 3 3 3 3 4 3 5 6 3 4 5 5 2 2 6 6 Midt 4 # 5 6 # 4 4 # 6 5 # 4 6 6 2 2 5 5 4 3 6 5 3 4 4 5 5 6 6 4 5 7 2 2 7 6 Lower 6 7 7 7 7 5 5 7 7 7 7 6 6 7 2 2 7 5 4 6 6 5 5 4 - - - - - - - - - - - - Balance - 7 - - 7 - - 7 - - 7 - - - - - - - - - - - - -Maybe you could hide a ring to your loved one, inside the "secret" hole. :-)
1. 2. 3 4 2 3 4 1 Top 4 4 2 4 4 1 4 1 1 4 2 2 3 5 2 3 5 1 Midt 3 5 5 3 5 5 6 6 1 6 6 2 3 5 2 3 5 # Lower 6 7 7 6 7 7 6 7 # 6 7 2 - - - - - - Balance - 7 - - 7 - - - - - - -However, there are no other solutions for this particular position of pieces 7 and 6 and,
When I have determined the solutions for: Case 1 Case 2 - - - - - - Top - - - - - - - - - - - - - - - - - - Midt 6 6 - - - - - - - 6 6 - 6 - - - - - Lower 6 7 7 6 7 7 - 7 - 6 7 - - - - - - - Balance - 7 - - 7 - - - - - - -Then I will have all solutions.
level: Top Midt Low Balance Solutions: 20 12 35 5 2 4 6 2 4 X 2 1 0 0 0 0 0 6 X X 0 X X X 0 - 0 0 3 0 0 0 0 23 X 9 0 0 0
level: Top Midt Low Balance - - - - - - - - - - - - - - - - - - 6 7 7 - 7 - - - - 6 6 - 6 7 - - - -So here are the number of solutions for each hole in Case 2:
level: Top Midt Low Balance Solutions: 20 10 50 4 0 2 6 0 0 8 21 10 0 0 0 2 0 3 2 2 0 X X X 0 - 0 4 5 0 X X 0 X X 11 0 0 0Again a total of 80 solutione.
- - - 6 7 7 6 7 -Due to pieces 6 7 - no piece can fall.
- 3 - 6 3 - 6 3 - 6 3 - 6 3 - - - -
- 3 - 5 3 - 5 3 - 5 3 - 5 3 - - 3 -In the image below you will find all the solutions with a gap in the lower level.