E-Mail.
|
|
SOMA News |
01 Dec 1998
E-Mail. |
The idea for the Checkered Soma was published in the first issue of The Soma Addict as an alternative way of forming the cube and other figures. This idea was created by Piet Hein himself and put down on paper on June 20, 1970. Shown below is a diagram depicting the proper placement of the checkered cubes. Try making one and see if you can build the cube for an added challenge!
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?!?
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 322Addendum:
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 453Now at 12.1.2000 My friend Courtney wrote to me:
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! Now at 26.8.2002 Steven Mai wrote to me:
/122/552/542 /116/376/544 /366/377/374 /322/552/542 /336/176/544 /366/177/174Now at 3.7.2006 Jim Waters wrote to me:
Standardization of solutions by bottom:
+---+---+---+
| | | |
+---+---+---+
Mdl | | 7 | |
+---+---+---+
| | | |
+---+---+---+
For reflections: For reflections:
+---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+
| | 7 | | | 4 | | | | | | 4 | | 2 | 2 | | | | 2 | 2 |
+---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+
Btms | | 7 | 7 | | 4 | 4 | | | | 4 | 4 | | | 2 | | | | 2 | |
+---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+
| | | | | | 4 | | | | 4 | | | | 2 | | | | 2 | |
+---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+
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
Tops 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
Mdls 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
Btms 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
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
Tops 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
Mdls 476 556 355 155 175 375 554 364 164 664 554 366 354
436 711 662 662 662 662 766 117 336 117 667 311 667
572 264 354 154 174 374 322 522 522 522 522 522 322
Btms 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
Interestingly enough, John Conway's solution which he thought was unique was the ONLY solution
which I had NOT already found! It is #13 on the chart. Note that Stuart Collins' cube is my #3;
the two ways found by Courtney are my #9 and #7 (see its reflection); and the two found by
Steven Mai are my #6 and #5 respectively.
When I first visited Science World BC in Vancouver BC Canada, they had a soma set and a pedestal
with the challenge and the claim that their exists only one solution, but without revealing what
it was; and I played with it and found my solution #1 (and notice piece 7 rests on the pedestal!!)
and I thought that was it. A assembled a soma cube of my own, rested it on a pedestal at home,
and challenged a friend of mine, Dan Seafeldt, who found what I list as solution #2. Since then,
I found the other solutions listed above.
The following figures are known as "couples." A couple is made with just one Soma Cube, but some of the seven pieces are independent of one another as the figures stand separately. Try to make the following couples and see what other ones you can come up with.
Do note that the shape #P002 is doubling the figure size, and that the figures #P001 and #P003 can be combined to form the cube.
Doubling each dimension of any piece will multiply the volume by 8. For any of the pieces comprised of 4 unit cubes, this increases the volume from 4 cubic units to 32 cubic units. For piece number 1, the volume increases from 3 cubic units to 24 cubic units. This equals the total number of unit cubes in the remaining 6 pieces. This poses a possible problem: Can you construct a figure similar to the number 1 piece but with all dimensions doubled, using the remaining 6 Soma pieces? The answer is yes, and the challenge is to construct this figure.