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Get one half by squaring (Posted on 2004-09-20) Difficulty: 4 of 5
Can you solve the following equation?

= 1/x + 1/y +...+ 1/z

All variables must be different, positive integers, and there must be a finite number of terms.

See The Solution Submitted by Federico Kereki    
Rating: 4.2500 (12 votes)

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The best that my program can do | Comment 14 of 31 |

A computer program comes up with the following variables:

2,3,4,5,6,11,54,519,59429

which totals about 0.4999999999999980848

and has some indication that starting an infinite series at 22686937 would help complete the total.

The program follows, and actually assumes that 2 will be one of the variables and starts from there without reporting the 2.

DECLARE SUB try (n#)
CLEAR , , 4000

DEFDBL A-Z
DIM SHARED lim, tot, used, remain, trm(50), howMany

pi = ATN(1) * 4
lim = pi * pi / 6 - 1.25
tot = 0
used = 0
remain = lim


try 3


SUB try (n)
 term = 1 / (n * n)
 rSave = remain
 remain = remain - term
 IF ABS(term + used - .25) < 1E-12 THEN
  FOR i = 1 TO howMany
    PRINT trm(i)
  NEXT
  PRINT n
 END IF
 IF term + used <= .25 + 1E-12 THEN
   uSave = used
   used = used + term
   howMany = howMany + 1
   trm(howMany) = n
   try n + 1
   howMany = howMany - 1
   used = uSave
 END IF
 IF remain >= .25 - used THEN
   rSave = remain
   uSave = used
   need = 1 / SQR(.25 - used)
   new = -INT(-need)
   FOR i = n + 1 TO new - 1
     term = 1 / (i * i)
     remain = remain - term
   NEXT i
   try new
   used = uSave
   remain = rSave
 END IF
 remain = rSave
END SUB

with output

3
4
5
6
11
54
519
59429
22686937
22686938
22686939
22686940
22686941
22686942
22686943
22686944
22686945
22686946
22686947
22686948
22686949
22686950
22686951
22686952
22686953
22686954
22686955
22686956
22686957
22686958
22686959
22686960
22686961
22686962
22686963


  Posted by Charlie on 2004-09-21 10:29:01
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