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 Sum of cubes and squares (Posted on 2011-08-28)
Consider N positive integers which are not necessarily distinct such that their sum of cubes is equal to 2011 (base ten) and their sum of squares is a perfect square.

Determine the smallest value of N for which this is possible. What is the next smallest value of N?

 No Solution Yet Submitted by K Sengupta No Rating

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 computer solution Comment 2 of 2 |

The smallest workable value of N is 10 and the next is 11.

As shown in the below first few lines of the sorted output of the program, N = 10 can be exemplified with three 3's, four 4's, a 6 and two 9's; as well as a 2, three 3's, three 4's, a 12 and two 1's; and by four 3's, two 4's, two 6's, a 7 and a 10.

The rows exemplifying N = 11 can be read in the same manner:

10          3  3  3  4  4  4  4  6  9  9 + 0 1's
10          2  3  3  3  4  4  4  12 + 2 1's
10          3  3  3  3  4  4  6  6  7  10 + 0 1's

11          2  2  2  3  3  3  3  7  8  8  8 + 0 1's
11          3  3  3  4  6  6  6  6  10 + 2 1's
11          2  4  4  4  5  5  6  7  10 + 2 1's
11          2  2  2  2  3  3  5  6  7  8  9 + 0 1's

Note that the program uses 1's as an afterthought and so appears as a count rather than as a 1 for each 1 making up the set, and that this count of ones is zero in a few instances.

Extended beyond these values:

10          2  3  3  3  4  4  4  12 + 2 1's
10          3  3  3  3  4  4  6  6  7  10 + 0 1's
10          3  3  3  4  4  4  4  6  9  9 + 0 1's

11          3  3  3  4  6  6  6  6  10 + 2 1's
11          2  2  2  2  3  3  5  6  7  8  9 + 0 1's
11          2  4  4  4  5  5  6  7  10 + 2 1's
11          2  2  2  3  3  3  3  7  8  8  8 + 0 1's

12          2  2  4  4  4  6  7  8  9 + 3 1's

13          3  3  5  5  5  5  5  5  6  6  6  6  7 + 0 1's
13          3  4  4  4  4  6  6  6  6  6  6  6  6 + 0 1's

14          3  3  3  3  4  4  4  4  4  5  5  11 + 2 1's

15          2  2  2  2  2  2  2  2  3  4  4  4  6  8  10 + 0 1's
15          2  2  2  2  2  2  3  3  3  3  3  3  7  9  9 + 0 1's

CLEAR , , 25000
DEFDBL A-Z

DIM SHARED cube(2 TO 12), square(2 TO 12)
DIM SHARED cuTot, sqTot, hist(260), numCt

FOR i = 2 TO 12
cube(i) = i * i * i
square(i) = i * i
NEXT

OPEN "sumcusq.txt" FOR OUTPUT AS #2

CLOSE

IF numCt = 0 THEN stNum = 2:  ELSE stNum = hist(numCt)
FOR addNum = stNum TO 12
cuTot = cuTot + cu: sqTot = sqTot + sq
IF cuTot <= 2011 THEN
numCt = numCt + 1
oneCt = 2011 - cuTot
sqTst = sqTot + oneCt
sr = INT(SQR(sqTst) + .5)
IF sr * sr = sqTst AND numCt + oneCt < 16 THEN
PRINT USING "#####"; numCt + oneCt;
PRINT ,
FOR i = 1 TO numCt
PRINT hist(i);
NEXT
PRINT "+"; oneCt; "1's"
PRINT #2, USING "#####"; numCt + oneCt;
PRINT #2, ,
FOR i = 1 TO numCt
PRINT #2, hist(i);
NEXT
PRINT #2, "+"; oneCt; "1's"
END IF
IF cuTot < 2011 THEN IF numCt < 16 THEN addOn
numCt = numCt - 1
END IF
cuTot = cuTot - cu: sqTot = sqTot - sq
END SUB

 Posted by Charlie on 2011-08-28 21:24:16

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