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Now, the Old Hat (Posted on 2009-02-20) Difficulty: 2 of 5
Solve this alphametic, where each capital letter in bold represents a different decimal digit from 0 to 9. None of the numbers can admit any leading zero.

                                 (NOW)THE = (OLD)HAT

See The Solution Submitted by K Sengupta    
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Solution computer solution | Comment 4 of 5 |

  10   '  nowtheldax  <--- string's assumed representation letters (x is not used)
  20   D="0123456789"
  30   H=D
  40   loop
  50    Ix=instr(D,"0")
  60    if Ix<3 or Ix=4 or Ix=5 then goto *NotThis
  70    Now=val(left(D,3))
  80    The=val(mid(D,4,3))
  90    Old=val(mid(D,2,1)+mid(D,7,2))
 100    Hat=val(mid(D,5,1)+mid(D,9,1)+mid(D,4,1))
 105    A=The*log(Now):B=Hat*log(Old)
 110    if abs(A-B)/B<0.000000001 then print Now;The;Old;Hat
 120   *NotThis
 130    gosub *Permute(&D)
 140    if D=H then goto 9999
 150   endloop
9999   end


729^680 = 243^816

(Check: 729 = 3^6 and 243 = 3^5, while 816/680 = 6/5.)

 

Added: going back to Daniel's comment of Mathematica taking under 10 minutes: the above UBASIC took a minute and a quarter to go through all possibilities. Note also, tolerance had to be allowed in the precision/accuracy when using logarithms, as exact equality shouldn't be expected when representations of transendental numbers are held in finite variable areas.

Edited on February 20, 2009, 12:50 pm
  Posted by Charlie on 2009-02-20 12:39:16

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