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Squaring the triangle (Posted on 2013-01-07) Difficulty: 3 of 5

The number 185136 has the interesting property that it is a triangular number that is also the product of 3 consecutive integers.

I square 185136 and each of the x consecutive numbers following it, and total the sum of all these squares, S.

Then I do the same with 185136+x+1 and the y consecutive numbers following it, until I again reach the same sum, S.

Find positive integer values of x,y,S compliant with the above requirements.

See The Solution Submitted by broll    
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Bit different with added consideration | Comment 4 of 7 |


While I concur with Charlie as to the Sum of Squares being 10471153462280 my program listing yields x=305 and y=303.

In fact x should be 304, and not because Charlie and broll say.

If I replace bs in the program listing below with a triangle number where n is even I can recreate the phenomenon.
Let bs = 3. The program reports x=2, y=0
 3 squared = 9 [after 1 step]
 4 squared = 16
Sum = 25
Next value is 5 and without and increment the square is 25.

Let bs = 10.  the program reports x=3, y=1.
After 2 steps:
10 squared = 100
11 squared = 121
12 squared = 144
Sum = 365
Next value is 13,  so
13 squared = 169
14 squared = 256
Sum = 365

My program is actually reporting the x+1 value.



CLS
DEFDBL A-Z
bs = 185136
s = 0: x = 0: eq = 0

DO
s = s + (bs + x) ^ 2
x = x + 1
y = 0: r = 0

DO
r = r + (bs + x + y) ^ 2
LOCATE 1, 1: PRINT s; r; x; y
 IF r = s THEN
  PRINT s, x, y
  eq = 1
 END IF
y = y + 1
LOOP WHILE r < s

LOOP WHILE eq <> 1
  Posted by brianjn on 2013-01-08 02:50:29
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