x, y and z (in this order), with x < y < z, are three positive integers in arithmetic sequence that satisfy this set of alphametic equations.

y⁴ - x⁴ = NEWS, and

z⁴ - y⁴ = LESS.

where, each of the capital letters in bold represents a different decimal (base 10) digit from 0 to 9.

Determine all possible value(s) of the triplet (x, y, z).

What are the possible four digit number(s) that represent WELL?

Note: Neither L nor N is zero.

The following program determines that z can be no larger than 14, as, if it were, then even only a 1 unit difference with y would make z^4-y^4 be longer than 4 digits:

list
   10   for Z=10 to 10000
   20   Diff=Z^4-(Z-1)^4
   25   print Z,Diff
   30   if Diff>9999 then print Z:end
   40   next
OK
run
 10      3439
 11      4641
 12      6095
 13      7825
 14      9855
 15      12209
 15
No for(next)

That leads to the following program and its run:

list
   10   for X=1 to 12
   20   for Y=X+1 to 13
   30   for Z=Y+1 to 14
   40      News=cutspc(str(Y^4-X^4))
   50      Less=cutspc(str(Z^4-Y^4))
   60      Good=1
   70      if len(News)<>4 or len(Less)<>4 then Good=0
   80      if Good then
   90        :for I=1 to 3
  100         :if instr(I+1,News,mid(News,I,1))>0 then Good=0:endif
  110        :next
  120        :if mid(Less,3,1)<>mid(Less,4,1) then Good=0:endif
  130        :if mid(Less,3,1)<>mid(News,4,1) then Good=0:endif
  140        :if mid(Less,2,1)<>mid(News,2,1) then Good=0:endif
  150        :if instr(2,Less,mid(Less,1,1))>0 then Good=0:endif
  160        :if instr(News,mid(Less,1,1))>0 then Good=0:endif
  170        :if Good then print X;Y;Z,News;" ";Less
  180   next
  190   next
  200   next
OK
run
 12  13  14     7825 9855
OK

So (x,y,z) can only be (12,13,14). WELL is 2899.

The program did not check for the integers x, y and z to be in arithmetic sequence, but the only set that fit the other criteria were in fact in arithmetic sequence.

In fact, even if we allowed x, y and z to be any digits between 0 and 14 without regard to which is largest or smallest, there are no further solutions.

Posted by Charlie on 2009-04-15 13:48:49