The problem statement from the Ole Miss Problems of the Week page is:

Find a two-digit number such that: the digits are different, the cube of the two-digit number is a five-digit number, all the digits of the five-digit cube are different, and all the digits of the five-digit cube are different from the original two-digit number.

Well, “find” has frequently indiciated to us that we need to write a calculator program to search for whatever is to be found. Should we do that in this case?

Of course we should! Bring out the CFX-9850G or FX-7400G, then!

Let us interpret the problem, first.
We want a two digit number for which the cube is a five digit
number.
That means the two digit number is between 22 (21^{3} is
only 9261) and 46 (47^{3} is 103823).
And we want all the digits of the number and its cube to be
distinct.
If we form a single number that is the number appended to its
cube, and then count the number of times each digit is used in
this constructed number, we can tell whether digits are
repeated.
If no digits are repeated in the constructed number, we have a
winner — I mean a solution to the problem.

That is enough of a description to start on the program. We will just loop through numbers between 22 and 46, inclusive, cube the number, append the original number to the cube, and count the incidence of digits. This program is available both here and as a text file with .CAT file contents. And, as always, semicolon (“;”) starts a comment not to be entered into the calculator.

Program P020600 (149 bytes):

; Variables: ; A is candidate number ; B is cube of candidate number ; C is candidate number appended to cube ; D is duplicate digits flag ; E is digit being examined ; F is count of appearances of digit being examined ; List 1[x+1] is count of digit x in C ; Symbols: ; -> is assignment arrow ; ^ is exponentiation operator ; <> is not equals relational ; / is division operator ; => is conditional jump operator ; _ is display triangle For 22->A To 46 ; Loop through all candidate numbers A^3->B ; Form the cube of the number 100B+A->C ; Append the original number to the cube Seq(0,D,1,10,1)->List 1 ; Zero a list to count digit incidence 0->D ; Zero a "duplicates" flag While C<>0 ; Loop through digits of combined number 10Frac (C/10)->E ; Extract the low order digit Int (C/10)->C ; Remove the low order digit E+1->E ; Add 1 to digit to get list element index List 1[E]->F ; Get previous digit count F+1->F ; Increment previous digit count F->List 1[E] ; Store new digit count F<>1=>1->D ; If new digit count > 1, digits are duplicated WhileEnd ; [End of loop through digits] D=0=>A_ ; If no digits are duplicated, display number Next ; [End of loop through candidate numbers] "Done" ; Display end message

Now we run this program, and it yields the output:

27 Done

As a check, we compute 27^{3}, which is 19683.
Indeed, all digits in the two numbers are distinct.

There’s probably an analytic way to do this problem, but I’m having a real difficult time getting motivated to figure it out.

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Copyright © 2001 Brian Hetrick

Page last updated 25 November 2001.