SOMA Crystal
SOMA News 27 Jan 2000

A new W-WALL impossibility proof:

!! Chun-Ju Lai's proof has been disputed by Bob Nungester,
Read his comments at the end of this letter.

I just received a short mail from Taiwan, Including a nice and simple proof that the W-Wall is still impossible.
Although structurally different, it actually resembles the original "Jotun's Proof", as it was made by Jotun Piet Hein.
Jotun's proof is working from the basis of checkerboarding a figure. (Described on page 38 of the Parker brothers manual [Newsletter 1998.12.10])
To see another description of checkerboarding, enter the main SOMA page, go to the SOLVER section, and then select SOLVER MANUAL. Here click on "How did we speed it extra up".

BUT - Here is the short proof.

Chun-Ju Lai This proof is by: Chun-Ju Lai (13 years old) junior high grade 1 student, and member of the "Chiu Chang Math. Club."
The words "Chiu Chang" means "Nine Chapters", in reference to an ancient Chinese scroll on mathematics.
Chun-Ju Lai lives in Hsin-Chu (70 KM far from Taipei, Taiwan).

<--- Chun-Ju Lai with some of his SOMA cubes

If we color the columns of the 'W-Wall' in an alternating fashion, starting by coloring the first column black. The total will be 15 black to 12 white.
Now, we examine each of the seven SOMA pieces, testing each in all possible orientations to ascertain the maximum number of black cubes it can have.
Because the pieces #5,#6,#7 are two levels in any direction, they must placed in a corner of the figure, except in first and last columns.
So, #5 must 1 black 3 white ;
#6 must 1 black 3 white ;
#7 must 2 black 2 white.
The remaining pieces are: then

         #1       #2       #3     #4   #5   #6   #7
Black   2(Max)   3(Max)   3(Max)   2    1    1    2
White   1(Min)   1(Min)   1(Min)   2    3    3    2

This chart show that the maximum number of black is 14,
NOT enough for Figure A033, which is thus impossible.

PS: The Math club is based in, Chiu Chang Math. Books & Puzzles Co. which owned by mr. Wen-Hsien SUN. and located at.
7Fl,No.5-1,Alley 15,Lane 147,Sec.3,Shin-Yi Road
Taipei,Taiwan,R.O.C. TEL:886-2-27048405 FAX:886-2-27067353
The compagny publishes only good books in mathematics, and imports meaningful mathematical games and puzzles into Taiwan.

DISPUTE: Chun-Ju Lai's proof has been disputed by Bob Nungester In this letter of 10. mar. 2000 where Bob writes:

I'm always interested in new ways to solve Soma figures, so I read Chun-Ju Lai's proof for the W-Wall to see if there was anything new.
The idea of identifying corners is interesting, but in analyzing the proof I don't think this particular argument works.
Pieces 5 and 6 do have to go in the corners, but they can be placed in a corner and have two black cubes and two white cubes (not 1 black, 3 white).
If you look at the orientation of piece 1 in his diagram and attach a fourth cube to the top cube in order to make piece 5 or 6, you will get an orientation with 2 black, 2 white which could be put in a corner.

Coloring the cubes this way, is another form of parity. we often refer to as Parity II.

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