Many mathematicians enjoy wordplay, and for obvious reasons. it is almost a branch of combinatorial mathematics. The pleasure derived from solving a combinatorial problem is very much like the pleasure of solving a cryptogram or a crossword puzzle, or constructing a good palindrome

Gardner, Martin

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MATHEMATICS AND W O W  PLAY MARTIN GARDNER (originally run February 1993) Many mathemat ic ians  enjoy w o r d p l a y ,  a n d  for  obvious  r e a s o n s .  I t  i s  almost a b r a n c h  of combina tor ia l  mathematic s .  The p l e a s u r e  d e r i v e d  from s o l v i n g  a  combina tor ia l  problem i s  v e r y  much l i k e  t h e  p l e a s u r e  of so lv ing  a  c ryp tog ram o r  a  c rossword  puzz le ,  o r  c o n s t r u c t i n g  a  good ps l ind rome .  Given t h e  formal  system of a r i t h -  met ic ,  anc i en t  mathema7icians a s k e d  themselves whe the r  t h e  d i g i t s  1  t h r o u g h  9 could  be p l aced  i n  a  three-by-three m a t r i x  so  t h a t  rows ,  columns,  a n d  the  two d i a g o n a l s  h a d  t h e  same sum. This  i s  not much d i f fe ren t  from a s k i n g  i f ,  g iven  the  formal  r u l e s  of E n g l i s h ,  one c a n  cons t ruc t  a  three-by-three word s q u a r e  i n  which e a c h  row,  column,  a n d  main d i a g o n a l  i s  a  d i f f e r en t  word.  I There  i s ,  of c o u r s e ,  a d i f fe rence  between combina to r i a l  mathemat- i c s  a n d  wordp lay .  Mathematics i s  embodied i n  t he  s t r u c t u r e  of t h e  u n i v e r s e .  Although mathemat ica l  systems a r e  f r ee  i nven t ions  of human minds ,  t hey  h a v e  a s t o n i s h i n g  a p p l i c a t i o n s  t o  n a t u r e .  No one expec ted  non-Eucl idian geometry to be u s e f u l ,  b u t  i t  p roved  to be e s s e n t i a l  t o  r e l a t i v i t y  t heo ry .  Boolean a l g e b r a  seemed use- l e s s  u n t i l  - s u r p r i s e !  - i t  t u r n e d  out to model t h e  e l e c t r i c a l  net- works of computers .  There  a r e  h u n d r e d s  of o t h e r  o u t s t a n d i n g  in-  s t a n c e s  of what  p h y s i c i s t  Eugene Wigner h a s  c a l l e d  t h e  "unreason-  a b l e  e f fec t iveness  of mathematics" .  Think  of t he  l e t t e r s  a n d  words of a l a n g u a g e ,  t oge the r  with i t s  r u l e s ,  a s  a  formal  system. Although the  words h a v e  a r b i t r a r y  mean ings  a s s i g n e d  to them b y  minds ,  a n d  t h e r e  may be a  "deep s t r u c t u r e "  of s y n t a x  t h a t  conforms to l o g i c ,  t he  words  themselves h a v e  no r e a l i t y  a p a r t  from a  c u l t u r e .  But te r f l ies  a r e  a l l  over t h e  wor ld ,  b u t  you wil l  not f ind  the  word "bu t t e r f ly"  by  looking t h r o u g h  a  teleicope o r  microscope. However, once t h e  word becomes a t t a c h e d  to b u t t e r f l i e s ,  i t  i s  amusing  to  obse rve  t h a t  bu t t e r f l i e s  f l u t t e r  b y .  Because l a n g u a g e ,  u n l i k e  mathemat ics ,  i s  " a r t i f i c i a l " ,  wordp lay  h a s  more in  common w i t h ,  s a y ,  i n v e n t i n g  c a r d  t r i c k s  o r  p l a y i n g  c h e s s .  The combina tor ia l  n a t u r e  of wordp lay  i s  unde r sco red  b y  t h e  r ecen t  u se  of computers  for  so lv ing  word problems.  Disks conta in-  i n g  a l l  t he  words of a  l a n g u a g e  a r e  now a v a i l a b l e .  With s u i t a b l e  p rog rams  they  c a n  be used to  cons t ruc t  word s q u a r e s ,  f i nd  a n a -  g r a m s ,  sho r t e s t  word l a d d e r s ,  a n d  so o n .  I w o u l d n ' t  b e  s u r p r i s e d  i f  some d a y  computers  wil l  solve complicated c rosswords  a s  e a s i l y  a s  t hey  now so lve  chess  problems.  I t  i s  worth no t ing  t h a t  both i n  mathematics  a n d  w o r d p l a y ,  solv- 212s u c h  conjec ture  1s curiously 11ke conflrrnlng a  t heo ry .  I n  so lv lng  for  example ,  one f l r s t  makes c o n ~ e c t u r e s .  I s  a  s ln-  A o r  I ,  o r  maybe O7 i s  ABCA the  word " tha t"?  s  a r e  then  tes ted  to see lf t hey  l e a d  to con t r ad lc -  t l o n s .  1fU they  l e a d  to  o the r  words ,  they  g a l n  In their p r o b a b l l l t y  of being co r rec t .  Even tua l ly  a  polnt  1s r eached  a t  whlch one  1s c e r t a l n  t h a t  a  c ryptogram h a s  been c racked  even though not  ,a l l  ~ t s  l e t t e r s  a r e  known. One i s  tempted to s a y  t h a t  when a l l  words a r e  known one c a n  be abso lu t e ly  c e r t a i n  a  c ryp tog ram h a s  been so lved .  Th i s  i s  not the  c a s e  because ,  e spec i a l ly  if t he  c ryptogram i s  s h o r t ,  t h e r e  j u s t  could be  ano the r  so lu t ion  t h a t  the  composer of t h e  puzzle  i n t e n d e d .  I f ,  however ,  t h e  c ryptogram i s  l o n g ,  such  u n c e r t a i n t y  becomes v a n i s h i n g l y  s m a l l .  This  i s  t r u e  a l s o  i n  sc ience .  When t h e r e  i s  a  l a r g e  a b u n d a n c e  of f a c t s  e x p l a i n e d  by  a  t h e o r y ,  such  a s  b y  the  Copernican theo ry  o r  t he  theory  of evolu t ion ,  c e r t a i n t y  r e a c h e s  a  p r o b a b i l i t y  of 0.99999999.. . Now for  a  deep me taphys i ca l  ques t ion .  If ches s  h a d  not  been i n v e n t e d ,  i s  t h e r e  a  s ense  i n  which theorems abou t  chess  c a n  be  s a i d  to e x i s t ?  Assuming the  formal  system of c h e s s ,  a n d  g iven  a  c e r t a i n  posi t ion on t h e  b o a r d ,  i s  i t  permiss ib le  to s a y  t h a t  t h e r e  i s  a  mate i n  t h ree  nioves even if no one h a s  posed t h e  prob-  - lem? Assuming the  s t r u c t u r e '  of a  deck of c a r d s ,  i s  t h e r e  a sense  i n  which a  good c a r d  t r i c k  i s  somehow "out t h e r e , "  i n  a  P l a ton ic  realm of u n i v e r s a l s ,  even if no c e r d s  ex i s t ed?  Suppose t h e r e  were no Eng l i sh  l a n g u a g e .  Would i t  be mean ing fu l  to s a y  t h a t  g iven  such  a l a n g u a g e ,  t h e r e  i s  a sense  i n  which a  c e r t a i n  a n a g r a m  "exis t s"  even  if no one spoke Eng l i sh?  I t  i s  something , l i ke  a s k i n g  i f  a  c e r t a i n  number with a  million d i g i t s  i s  p r ime o r  composite before anyone  %as  t e s t e d  the  number t o  f i n d  ou t .  Well, not q u i t e ,  because  a ~ . i t h m e t i c  c e r t a i n l y  "ex is t s"  a s  a  formal  sys tem.  Anyway,  most mathemat icans  a r e  P l a ton i s t5  who be l ieve  t h a t ,  no mat te r  how b i z a r r e ,  o r  how f a r  removed from r e a l i t y  a  system c a n  b e ,  t h e y  "discover"  i t s  theorems r a t h e r  t h a n  inven t  them. Even though Eng l i sh  i s  a  human cons t ruc t ion ,  nowhere to  be found i n  n a t u r e ,  i s  t h e r e  a  sense  i n  which i t s  w o r d p l a y s  a r e  " r ea l "  before anyone  f i n d s  them? I l e a v e  answer ing  t h i s  t o  my r e a d e r s .  213

Mathematics and Word Play

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Mathematics and Word Play

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