SOMA News 8 Mar 1999 E-Mail.

### PROOF THAT SOMA FIGURE 029 IS IMPOSSIBLE:

Robert Staatz 03868300193-0001@t-online.de (From Germany) proves why the figure 029 is impossible.
The original text was in German, look inside the html code of this page, to find the german text at the end.

When working with the SOMA 029 I had the idea to leave one of the pieces out, and it seem's that all three figures SOMA 027, 028 and 029 can be made in this way, except - they all miss one level.

Proving the 029 Impossibility

Not all the possible combinations of the pieces are shown. On most of the drawings I have shown the figure 029 partially so that it is easier to see.

Piece no. 2 can ONLY be placed at the top of the figure.

This picture show that it is not possible for piece no 4 to lay flat at the lowest layer, because in one of the corners, no further piece can be placed.
In Number 1 we could also use piece 5 instead of piece 1.
In stead of piece 5 in figures 2 and 3, we could also use the other pieces, except piece 2.
The front position in number 4, cannot be used by any other piece. The pieces 1 and 5 are shown only as examples.

This picture show that it is also not possible when piece 4 is NOT lying flat on the bottom level In number 1 to 4, we see piece 4 at one side, and even though number 3 looks as if we could solve SOMA 029, it is unfortunately impossible to fit piece 7.
Also here, the front part can only be solved with piece 2, and that is in use at the top.

Even the piece 7 will not find place in the lowest level, as shown on the drawing.
If piece 7 is placed at the rear corner (Numbers 1 through 4) then we may only use one more piece.
In numbers 5 through 8, the piece 7 is to one side. Here it is only possible to fit piece 1 in the front.
No piece may in number 9 be placed in the front position.
Because piece 7 is covering a hole in the lowest level in number 10.

In this drawing, the SOMA 029 is buildt from the top down, If the piece 7 is placed with piece 2, as in number 1, then we immediately get a hole that we cannot fill.
In number 2 the pieces 1 and 7 fit nicely with piece 2 but piece 4 will block any further placements. Also, piece 3 cannot be used with piece 1 or 4 to align with piece 2.
With the pieces 1, 2, 4 and 7 you may think you have made it, (as in number 5 and 6) but then read the next paragraph.

This drawing show all the possible combinations using pieces 3, 5 and 6.
Unfortunately, it is not possible to combine the 5 and 6 from the previous image, with these shapes, in order to make SOMA 029.
Remaining is only the figures SOMA 027 and SOMA 028.

- Proved and submitted by Robert Staatz.

# Here is the same proof in text notation

NOTE: the letter '#' mean that at this spot no further piece can be placed at this spot.
The letter '*' mean that other pieces can be placed at this spot.

This presentation show only the lowest level. the rest of the buildup will be at the upper left corner.

Even if the lowest level should be complete, will the rest of the figure be impossible to solve, (See the description for pieces 4 and 7).

Piece 2 must be at the top

Piece 4 cannot be flat at the bottom level:

```   | 4 1 1 | 5 4 # | 5 4 # | 5 5 4 |
| 4 4 1 | 4 4 1 | 1 4 4 | 1 4 4 |
| # 4 # | 4 1 1 | 1 1 4 | 1 4 # |```

Piece 4 cannot be not flat at the bottom level:

```   | 5 5 # | 6 6 # | 3 3 3 | 3 3 3 | 3 1 1 |
| 4 1 # | 4 6 1 | 4 3 1 | 4 5 5 | 3 4 1 |
| 4 1 1 | 4 1 1 | 4 1 1 | 4 # 5 | 3 4 # |```

Piece 7 cannot be at the bottom level:

```   | 7 7 1 | 7 7 3 | 7 7 # | 7 7 6 |
| 7 1 1 | 7 3 3 | 7 5 # | 7 6 6 |
| # # # | # # 3 | 5 5 # | # # # |

| 3 7 7 | # 7 7 | # 7 7 | 6 7 7 |
| 3 7 1 | 4 7 1 | 5 7 1 | 6 7 1 |
| 3 1 1 | 4 1 1 | 5 1 1 | # 1 1 |

| 3 1 1 | 7 # 3 |
| 3 7 7 | 1 3 3 |
| 3 7 # | 1 1 3 |```

The proof for the levels 2 to 7 (to show the complete figure):

```   | 2 . .   2 . .   2 2 .   7 * .   * * .   * * .   * * * |
| . . .   . . .   7 # .   7 7 .   * * .   * * .   * * * |
| . . .   . . .   . . .   . . .   . . .   . . .   * * * |

| 2 . .   2 . .   2 2 .   7 7 .   # 7 .   * 4 .   * 4 * |
| . . .   . . .   1 1 .   1 7 .   # 4 .   * 4 .   * * * |
| . . .   . . .   . . .   . . .   . . .   . . .   * * * |

| 2 . .   2 . .   2 2 .   1 3 .   # # .   * * .   * * * |
| . . .   . . .   1 3 .   1 3 .   4 3 .   4 4 .   * 4 * |
| . . .   . . .   . . .   . . .   . . .   . . .   * * * |

| 2 . .   2 . .   2 2 .   3 4 .   1 4 .   1 * .   * * * |
| . . .   . . .   3 4 .   3 4 .   3 # .   1 * .   * * * |
| . . .   . . .   . . .   . . .   . . .   . . .   * * * |

| 2 . .   2 . .   2 2 .   1 4 .   7 4 .   * * .   * * * |
| . . .   . . .   1 4 .   1 4 .   7 7 .   7 * .   * * * |
| . . .   . . .   . . .   . . .   . . .   . . .   * * * |

| 2 . .   2 . .   2 2 .   4 1 .   4 7 .   * * .   * * * |
| . . .   . . .   4 1 .   4 1 .   7 7 .   * 7 .   * * * |
| . . .   . . .   . . .   . . .   . . .   . . .   * * * |```

This leaves only the pieces 3, 5 and 6 to make the lowest level. Using these pieces there are 4 possible solutions for the lowest level:

```   | 3 5 5 | 3 3 3 | 6 5 5 | 5 5 3 |
| 3 6 5 | 6 5 5 | 6 3 5 | 6 3 3 |
| 3 6 6 | 6 6 5 | 3 3 3 | 6 6 3 |```

From the four solutions of the lowest level, we only get the figures 027 and 028.

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