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ten*ten ten = ninety (Posted on 2009-10-25) Difficulty: 2 of 5
In this alphametic equation, each of the small letters in bold denotes a different base b digit from 0 to b-1. Neither t nor n can be zero.

(ten)*(ten) - ten = ninety

Determine the minimum value of b, for which there exists at least one solution to the above equation.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Solution Next larger solution Comment 5 of 5 |

The next base with a solution is base 66.  T=60, E=51, N=55, I=64 and Y=0.  In base 10, TEN=264781 and NINETY=70108713180.

I used the following in UBASIC, and weeded out a few bad solutions as I did not put in vigorous checking.

   10   B_min=5
   11   B_max=11
   12   for B=B_min to B_max
   20   for T=ceil(sqrt(B)) to B-1
   30   N_min=floor(T*T/B)
   31   for N=N_min to N_min+2
   32   if N=B then 97
   33   if N=T then 97
   39   ' print B,T,N
   40   for E=0 to B-1
   41   if E=N then 96
   42   if E=T then 96
   50   Ten=B*(B*T+E)+N
   51   Noneto=B*(B*(B*(B*B+1)*N+E)+T)
   52   Ioooy=Ten*(Ten-1)-Noneto
   55   Y=Ioooy@B
   56   I=floor(Ioooy/(B*B*B*B))
   60   Zero=Ioooy-Y-I*B*B*B*B
   70   if Zero=0 then print B,N;E;T;I;Y,Ten;Noneto+Ioooy
   96   next E
   97   next N
   98   next T
   99   next B

  Posted by Brian Smith on 2009-10-30 22:59:24
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