Every day, Lucas receives a cube of metal whose side is between 9.5 and 10.5 meters. He melts the cube down and uses the resulting metal to make smaller cubes, whose sides are all between 9.5 and 10.5 centimeters, leaving no metal left over.

One day Lucas determined that he had made the smallest number of cubes he could under these conditions. It was a 6-digit number, call it abcdef, but not necessarily implying that there are no duplicate digits.

Lucas decided to make a cryptarithm using his number:

 RSTSTS
+RTRUTT
 abcdef

where each of the capital letters represents a distinct digit, the same digit for each occurrence of that letter.

What is RUST?

The quantity of smaller cubes is minimized when their size is maximized while the size of the large cube is minimized.  If the sides of the small cube are set at 10.5 cm while the sides of the large cube are set at 950 cm, the number of smaller cubes derived is:

(950^3)/(9.5^3) = 740632.761 . . .

Therefore, the smallest quantity of smaller cubes that can be made is 740633.

 RSTSTS
+RTRUTT
 740633

R must equal 3, and S+T must be greater than 10, so S+T=13.  T must then equal 6 since T+T+1=3 or 13 and T>4.  This means S=7 and U = 8.

 376767
+363866
 740633

RUST = 3876

Posted by hoodat on 2007-02-23 15:47:48