SOMA News 1 Feb 1999 E-Mail.

# Winning Ways for your mathematical plays

Volume 2: Games in particular
Elwyn R. Berlekamp, John H. Conway, Richard K. Guy

E-Mail to Mr. John H. conway@math.Princeton.EDU

## SOMA

This elegant little puzzle was devised by Piet Hein. Figure 1 shows the seven non-convex shapes that can be made by sticking 4 or fewer 1 x 1 x 1 cubes together. Piet Hein's puzzle is to assemble these as a 3 x 3 x 3 cube.

1=W 2=Y 3=G 4=O 5=L 6=R 7=B

White Yellow Green Orange bLue Red Black
Figure 1. The Seven Pieces of Soma.

We advise you to use different colors for your pieces as in the figure. Many people solve this puzzle in under ten minutes, so it can't be terrible hard. But we've got a distinct feeling that it's much harder than it ought to be. Is this just because the pieces have such awkwardly wriggly shapes?

THE HIDDEN SECRETS OF SOMA
It's because the Soma puzzle pieces have to satisfy some hidden constraints as well as the obvious ones, that it causes most people more trouble than it should. Let's see why.
The 3 x 3 x 3 cube has 8 vertex cells, 12 edge cells, 6 face cells and 1 central cell as in fig. 2.

Figure 2. The Vertex, Edge, Face and Central (invisible) Cells.

Now the respective pieces can occupy at most

```W  Y  G  O  L  R  B
1, 2, 2, 1, 1, 1, 1```
of the vertex cells, so just one piece, the deficient one, must occupy just one less vertex-cell than it might. The green piece can't be deficient without being doubly so, and therefore:

 the Green piece has its spine along an edge of the cube.

Now let's color the 27 cells of the cube in two alternating colors,

Flame for the 14 FaVored cells, F and V,
Emerald for the 13 ExCeeded ones, E and C.

Then in onesolution that we know, the respective pieces occupy

```W   Y   G   O   L   R   B
2 + 2 + 3 + 2 + 2 + 2 + 1 = 14 F,V cells,
1 + 2 + 1 + 2 + 2 + 2 + 3 = 13 E,C cells.```
but the Yellow, Orange, bLue and Red pieces, and we now know also the Green piece, must occupy these numbers in every solution, and therefore so must the White and the Black, since an interchange of colors in either or both of these would alter the totals.

 The White piece occupies 2 FV cells, 1 EC cell.

 The Black piece occupies 1 FV cell and 3 EC ones.

For the placing of a single piece within the box, these considerations leave only the positions of Fig. 3 (below). You'll see that up to rotations of the cube, the placement of any single piece is determined by whether or not it is deficient and whether or not it occupies the central cell.

The hidden secrets of Soma make it quite likely that one of the first few pieces you put in may already be wrong, when of course you'll spend a lot of time assembling more pieces before such a mistake shows its effect. This would happen for instance if you started by putting the corner of the White piece into a corner of the cube. But if you only put the pieces into the allowed positions, you'll find a solution almost as soon as you start.

Fig. 3. All possible Positions for the Seven Soma Pieces.

The complete list of 240 Soma solutions was made by hand by J.H.Conway and M.J.T.Guy one particularly rainy afternoon in 1961.

solutions to SOMA Cube (1=W 2=Y 3=G 4=O 5=L 6=R 7=B) are shown in 3 slices:
```BBL Top slice
BOO
OOY

BWL Middle slice
RLL
RRY

GWW Bottom slice
GGY
GRY
----------------------------------------------------------- 001..015
BBL BBW BBR BBR BBW BBW BBW BBL BBW BBR BBR BBR BBW BBL BBW
BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO
OOY OOY OOY OOW OOR OOY OOY OOY OOY OOY OOY OOW OOR OOY OOY

BWL BLW BRR BRR BWW BRW BLW BWL BLW BRR BRR BRR BWW BGL BGW
RLL GRW GRL GRW GRR GRW GRW RLL RLW LRG WRL LRW LRR RLL RLW
RRY YYY YYY LLW LLR LLY RRY YYY YYY YYY YYY LLW LLR YYY YYY

GWW GLL GWW GYY GYY GRR GLL WWG LLG LLG GGG GGG GGG GGG GGG
GGY GRL GWL GLY GLY GLY GRL RGG RGG LWG WGL LGY LGY RWW RLL
GRY GRR GLL GLY GLY GLY GYY RRG RRG WWG WLL YYY YYY RRW RRL
----------------------------------------------------------- 016..030
BBL BBW BBL BBL BBL BBR BBW BBR BBW BBY BBY BBY BBG BBG BBG
BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO
OOW OOL OOY OOY OOY OOY OOY OOW OOR OOR OOW OOW OOY OOY OOW

BGL BGW BGL BGL BGL BRR BLW BRR BWW BWY BRY BLY BLG BWG BRG
RLL RLW RLL RLL WLL LRW RRW LRW LRR LRR LLW RLW RLG RLG LLG
RWW RLL RRY RWY WRY YYY YYY LLW LLR LLR LGW RGW YYY YYY LWW

GGG GGG GGG GGG GGG LLW RLL YYY YYY WWY RRY LLY LLG WWG RRG
RRY RRY WWY RRY RRY LGW RGL LGY LGY LGY RLY RRY RWW RLL RLY
YYY YYY WRY WWY WRY GGG GGG GGG GGG GGG GGG GGG RRW RRL YYY
----------------------------------------------------------- 031..045
BBG BBG BBG BBG BBG BBG BBG BBG BBG BBG BBG BBG BBG BBG BBG
BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO
OOL OOW OOL OOY OOY OOY OOY OOY OOY OOY OOY OOY OOW OOW OOW

BWG BLG BRG BRG BLG BLG BLG BRG BRG BRG BGG BGG BGG BGG BGG
RLG RLG WLG LLG RLG RLG WLG LWG WLG WLG LRR WRL LLW RLW LLW
RLL RWW YLL LWY RRY RWY WRY LWY WWY LLY YYY YYY LRW RRW YRW

WWG LLG WRG RRG LLG LLG LLG RRG RRG WRG LLG RRG YYG YYG YLG
RRY RRY WRR RLY WWY RRY RRY RWY RLY WRR LWR WRL YLR YLL YLR
YYY YYY YYY WWY WRY WWY WRY LLY LLY LYY WWR WLL YRR YRL YRR
----------------------------------------------------------- 046..060
BBG BBG BBG BBG BBG BBG BBG BBR BBW BBW BBR BBR BBR BBY BBY
BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO
OOR OOW OOW OOW OOW OOY OOY OOG OOG OOG OOG OOG OOG OOG OOG

BGG BGG BGG BGG BGG BGG BGG BRR BWW BWW BRR BRR BRR BRY BLY
LRR LRW LRW LRW RRL LWW RRL WRG RRG LRG LRG WRG LRG LLG RLG
LLR LLW RRW YRW YWW RWY WWY WLG RLG LRG LLG LLG WWG LWG RRG

WWG RRG LLG LLG RLG LLG RLG YYY YYY YYY YYY YYY YYY RRY LLY
LWY LRY LRY LRR RLL LRY RLL YLL YLL YRR LWY WLY LLY RLY WWY
YYY YYY YYY YYY YYY RRY WYY WLG RLG LLG WWG WLG WLG WWG WRG
----------------------------------------------------------- 061..075
BBY BBY BBY BBY BBW BBW BBR BBW BBW BBW BBY BBY BBY BBY BBY
BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO
OOG OOG OOG OOG OOG OOG OOG OOG OOG OOG OOG OOG OOG OOG OOG

BLY BLY BRY BRY BWW BLW BRR BLW BRW BRW BRY BLY BWY BRY BRY
RLG WLG LWG WLG LLR RRW LRW LLW LRW LRW WLL WLL LWW LWW LRR
RWG WRG LWG WWG LGG YGG LGG RGG LGG LGG WGG RGG RGG LGG LGG

LLY LLY RRY RRY YRR YLL YYY YYY YYY YRR RRY WLY LLY RRY WYY
RRY RRY RWY RLY YLR YRL YWW LRY RRY YYY RLY WRY LRY RWY WWR
WWG WRG LLG LLG YYG YRG LLG RRG LLG LLG WLG RRG RRG LLG LLG
----------------------------------------------------------- 076..090
BBG BBG BBG BBG BBG BBW BBW BBW BBW BBY BBY BBW BBW BBY BBL
BGG BGG BLG BWG BWG BGW BGW BWW BWW BLY BWY BWW BWW BLY BGL
WWG WWG LLG WWG WWG GGG GGG GGG GGG LLY WWY YYY YLL LLY GGG

BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BLL
OOR OOR OOG OOG OOG OOW OOW OOR OOR OOR OOL OOL OOL OOR ROO
YWL LWR LWR RLL RRL RLL RRL YGL LGR LWY RRY RRY YRL LGY OOY

YRR YYY YYY YYY YYY YYY YYY YRR YYY GRR GGG GGG GGG WRR WWY
YLR LLY YRR YRL RLY YRL RLY YLR LLY GGR RGL RGL YGR WWR RWY
YLL LRR WWR RRL RLL RRL RLL YLL LRR GWW RLL RLL YRR GGG RRY
----------------------------------------------------------- 091..105
BBR BBL BBY BBW BBR BBG BBG BBG BBG BBL BBL BBL BBW BBR BBY
BRR BWL BWY BWW BRR BLG BLG BLG BWG BGL BGL BGL BGW BRR BLY
YYY WWY WWY YYY YYY LLG LLG LLG WWG GGG GGG GGG GGG YYY LLY

BLR BLL BRR BRR BLR BRR BRY BRR BRL BLL BLL BLL BRL BWR BRY
GOO ROO LOO LOO WOO OOG OOG OOG OOG OOR OOW OOR OOW OOL OOG
OOY OOY OOY OOY OOY LOO LOO LOO YOO YOO ROO YOO YOO YOO LOO

GLL GGG LLR LLR WLL YYR RRY WWR RRL YRR YYY WRR RRL GWW RRG
GWL RGY LGR LGR WGL YWR RWY YWR RLL YWR YRW WWR RLL GGL RWG
GWW RRY GGG GGG GGG YWW WWY YYY YYY YWW RRW YYY YYY GLL WWG
----------------------------------------------------------- 106..120
BBR BBY BBY BBG BBG BBG BBG BBG BBG BBW BBR BBY BBG BBG BBG
BRR BLY BLW BGG BGG BGG BWG BLG BRG BGW BGR BGY BLR BWR BWY
YYY LLY LLW RRG WRG LRG WWG LLG RRG GGG GGG GGG LLR WWR RRY

BGR BRY BRY BLL BLL BWW BLL BWW BWW BLL BWW BLY BWG BLG BWG
OOL OOW OOW RLW WLR LLR RLG RRG LRG RLW LRR LLR WWG LLG RWG
YOO LOO LOO RWW WRR LRR YYY LRY LLY YYY LLY WWR LRR YRR RLY

GGG RRW RRY OOL OOL OOW OOL OOW OOW OOL OOW OOY OOG OOG OOG
WWL RGW RGY YOO YOO YOO ROO ROO LOO ROO LOO LOO YOO LOO LOO
WLL GGG GGG YYY YYY YYY RRY YYY YYY RRY YYY WRR YYY YYY LLY
----------------------------------------------------------- 121..135
BBG BBG BBG BBG BBG BBG BBG BBG BBG BBG BBG BBG BBG BBG BBG
BRR BLY BLY BRY BLL BLL BRR BRR BWL BLL BLL BRR BRR BWW BWW
WYY YYY YYY YYY YYY YYY YYY YYY WWL RRL RRW LRW LWW LRW RRL

BRG BGG BGG BGG BGG BGG BGG BGG BGG BGG BGG BGG BGG BGG BGG
WRG RLW WLR RRW RLW WLR LLR WLR RLL RWW RLW LLW LLR LLR RLW
WLY LLW LLR RLW YLW YLR YWR YLR YYY RWL RLW LRW LWR LRR RLL

OOG OOG OOG OOG OOG OOG OOG OOG OOG OOG OOG OOG OOG OOG OOG
LOO ROO WOO LOO ROO WOO LOO WOO ROO YOO YOO YOO YOO YOO YOO
LLY RRW WRR LLW RRW WRR LWW WLL RRY YYY YYY YYY YYY YYY YYY
----------------------------------------------------------- 136..150
BBG BBG BBG BBG BBG BBG BBG BBG BBG BBY BBY BBY BBG BBG BBR
BLW BRW BWL BLY BLY BWY BRY BLL BLR BWY BWY BLY BRG BLG BGR
LLW RRW YLL LLY LLY RRY WRY RLY LLY GWW RRG LLG RRG RRG GGG

BGG BGG BGG BGG BGG BGG BGG BGG BGG BLY BWY BWY BWW BLL BWW
RRW LRW WWR WRR RWW RWW WRR WRL WRR LLR RWG RWW LRG RLG LRR
LRY LLY YLR LRY LWY RLY WLY RRY LRY GGR RLG LGG YYY RWW YYY

OOG OOG OOG OOG OOG OOG OOG OOG OOG OOY OOY OOY LLW YYY LLW
ROO LOO YOO WOO ROO LOO LOO WOO WOO LOO LOO ROO LOO YOO LOO
YYY YYY YRR WRY RRY LLY LLY WYY WYY GRR LLG RRG OOY OOW OOY
----------------------------------------------------------- 151..165
BBY BBL BBY BBY BBW BBW BBG BBG BBG BBG BBG BBG BBG BBG BBG
BGY BRL BLY BRY BLY BRY BRR BLY BLY BLY BWY BRY BLL BRR BWW
GGG GGG LLY RRY LLY RRY YYY LLY LLY WLY LWY RRY WYY WYY YYY

BRY BLL BGY BGY BGW BGW BRG BWG BRG BRG BRG BWG BRG BRG BGG
WLL RRW RRW LRW RRW LRW LRG RRG WWG LLG LLG LRG WLG LRG LRW
WWL RGW LRW LLW LRY LLY YWW LRY LWY WWY LWY LLY WLY WWY YRR

RRY YYY GGG GGG GGG GGG LLG WWG RRG RRG RRG WWG RRG LLG LLG
ROO YOO ROO LOO ROO LOO LOO ROO ROO ROO ROO LOO ROO LOO LOO
OOL OOW OOW OOW OOY OOY OOW OOY OOY OOY OOY OOY OOY OOY OOR
----------------------------------------------------------- 166..180
BBG BBG BBG BBW BBL BBL BBL BBW BBY BBY BBY BBY BBY BBR BBL
BRW BRY BWW BLL BRL BWL BRL BRR BLY BLY BLY BWY BRY BLL BLL
RRW WRY RYY RRG RRG RRG WWG LRG LLG LLG WLG LWG RRG WLG RRG

BGG BGG BGG BWW BLL BLL BLL BWW BWY BRY BRY BRY BWY BRR BWW
LRW LRR LRW RLG WRG RWG RRG LLG RRG WWG LLG LLG LRG WRL RLW
YYY WWY RRY RLG WWG RWG RWG LRG LRG LWG WWG LWG LLG WGG RGG

LLG LLG LLG YYY YYY YYY YYY YYY WWY RRY RRY RRY WWY YYY YYY
LOO LOO LOO YOO YOO YOO YOO YOO ROO ROO ROO ROO LOO YOO YOO
OOY OOY OOY OOG OOG OOG OOG OOG OOG OOG OOG OOG OOG OOG OOG
----------------------------------------------------------- 181..195
BBL BBL BBW BBL BBR BBW BBY BBY BBY BBR BBR BBL BBL BBL BBR
BWL BRL BLW BGL BGL BGL BGY BGL BRR BRR BLL BRL BRL BWL BLW
RRG WRG RRG GGG GGG GGG GGG GGG GGG YYY GLW GRR GRW GWR GLW

BLL BLL BLL BLL BRR BWW BLY BWY BLY BGR BRR BLL BLL BLL BRR
RWW WRR RLW RRW WRL RRL WLR RRL WLR WLL GRL GWW GRR GRR GRW
RGG WGG RGG RWW WLL RLL WRR RLL WGR WLY GWW GWR GWW GWR GLL

YYY YYY YYY YYY YYY YYY LLY WWY LLY GGG YYY YYY YYY YYY YYY
YOO YOO YOO OOY OOY OOY OOR OOY OOY OOL OOY OOY OOY OOY OOY
OOG OOG OOG ROO WOO ROO WOO ROO WOO WOO GOO GOO GOO GOO GOO
----------------------------------------------------------- 196..210
BBR BBY BBY BBY BBY BBY BBY BBR YYY WYY RYY WYY RYY RRY GRR
BWL BLY BRY BWY BRR BRR BWW BRR LBB LBB RBB LBB RBB RBB YBB
GWW GLL GRR GWW GRW GWW GRW GLL LBG LBG WBG LBG WBG WBG YBW

BRR BWY BWY BLY BLY BLY BLY BWR YWW WWY RRY WRY RRY LLY GLR
GRL GLR GLL GLR GLW GLR GLR WWL RBG RBG WBG WBG LBG RBG GBR
GLL GRR GLR GRR GRW GWR GRR GGL LLG LLG WLG LLG WWG WWG YWW

YYY WWY WWY LLY LLY LLY LLY YYY OOW OOY OOY RRY LLY LYY GLL
OOY OOR OOL OOR OOY OOY OOY OOY ROO ROO LOO ROO LOO LOO OOL
GOO GOO GOO GOO GOO GOO GOO GOO RRG RRG LLG OOG OOG OOG YOO
----------------------------------------------------------- 211..225
YRR LLY GOO GOO YOO YYY YYY GOO OOG YRR YYY YYY YRR LRR YLL
YBB LBB OOY OOL OOL YOO YOO OOY ROO YOO YOO YOO YOO LLR YRL
GBW GBW YYY YLL GLL OOW OOG YYY RRW OOW OOW OOG OOG GGG YRR

YLR RLY GBB GBB YBB RBB RBB GGL LGG YLR LRR LRR YLR LRW YWW
GBR RBW GBW GBR GBR LBG LBW BBL BBW BBR BBW BBG BBR BBW BBL
GWW GGW RRL YLR GLR LWW LGG BWR BRW BWW BGW BWG BGG BGY BGR

YLL RRY GBW GBW YBW RBG RBW GLL LLG YLL LLR LLR YLL OOW OOW
OOL OOY RLW YWW YWW RRG RRW BRR BLY BGL BLR BLR BWL BOO BOO
GOO GOO RLL YRR GRR LLG LLG WWR YYY GGG GGG WWG WWG YYY GGG
----------------------------------------------------------- 226..240
LRR GGG RLL YRR LLR LRR YLL LRR LRR GWW LLR WWL LRR RLL YRR
LLR RGY RGL YGR LRR LLR YLR LLY LLY RWY LRR RWL LLR RWL YWR
YYY RRY GGG GGG GGG GGG YRR YYY WWY RRY WWG RRG WWG WWG WWG

LRW WWL RRY YRL YLR LRY YGL LGR LGR GGL YLR YLL LRY RRY YRL
BBW BBL BBL BBL BBW BBW BBR BBR BBR BBL BBG BBG BBG BBL BBL
BGY BRY BWW BWW BGW BGW BWW BWW BWY BRY BWG BRG BWG BGG BGG

OOW WLL YYY YLL YYY YYY GGG GGG GGG GLL YYY YYY YYY YYY YLL
BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO BOO
GGG OOY OOW OOW OOW OOW OOW OOW OOY OOY OOG OOG OOG OOG OOG
-----------------------------------------------------------
```