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A Pythagorean triplet (Posted on 2012-08-28) Difficulty: 3 of 5
Given that a2+b2 = c2
find the value of a*b*c if a+b+c=1000.

See The Solution Submitted by Ady TZIDON    
Rating: 5.0000 (2 votes)

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Solution computer solution | Comment 2 of 4 |

Taking into consideration the title, a pythagorean triple should consist solely of positive integers and the answer is  a=200, b=375, c=425; a*b*c=31,875,000.

However, based on the puzzle text, the following program also allows for negative integers:

DEFDBL A-Z
CLS
FOR tot = 2 TO 99999
  FOR a = 1 TO tot / 2
     b = tot - a
     csq = a * a + b * b
     c = INT(SQR(csq) + .5)
     IF c * c = csq THEN
       t = a + b + c
       IF t >= 1000 THEN
         IF t = 1000 THEN PRINT a; b; c, a * b * c
         IF -a + b + c = 1000 THEN PRINT -a; b; c, -a * b * c
         IF a - b + c = 1000 THEN PRINT a; -b; c, -a * b * c
         IF a + b - c = 1000 THEN PRINT a; b; -c, -a * b * c
         IF a - b - c = 1000 THEN PRINT a; -b; -c, a * b * c
         IF -a + b - c = 1000 THEN PRINT -a; b; -c, a * b * c
         IF -a - b + c = 1000 THEN PRINT -a; -b; c, a * b * c
       END IF
     END IF
  NEXT
NEXT

and finds, up to the point where manually stopped:

  a    b    c                 a*b*c
 200  375  425               31875000
-250  600  650              -97500000
 750 -1000  1250            -937500000
 800 -1500  1700            -2040000000
 840 -2125  2285            -4078725000
 1625  1800 -2425           -7093125000
 1500  2000 -2500           -7500000000
 1400  2250 -2650           -8347500000
 875 -3000  3125            -8203125000
 1250  3000 -3250           -12187500000
 1200  3500 -3700           -15540000000
 900 -4000  4100            -14760000000
 1160  4125 -4285           -20503725000
 1125  5000 -5125           -28828125000
 920 -5250  5330            -25743900000
 1100  6000 -6100           -40260000000
 1080  7250 -7330           -57393900000
 950 -9000  9050            -77377500000
 1050  11000 -11050         -127627500000
 960 -11500  11540          -127401600000
 1040  13500 -13540         -190101600000
 968 -14625  14657          -207499149000
 1032  16625 -16657         -285784149000
 975 -19000  19025          -352438125000
 1025  21000 -21025         -452563125000
 980 -24000  24020          -564950400000
 1020  26000 -26020         -690050400000
 984 -30250  30266          -900897756000
 1016  32250 -32266         -1057227756000

But clearly there are no further all-positive solutions, as we've reached the point where a+b alone equals 33266, much more than 1000.


  Posted by Charlie on 2012-08-28 11:14:42
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