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Square Egyptians (Posted on 2017-06-28) Difficulty: 3 of 5
Express 1/2 as the sum of Square Egyptian Numbers with distinct denominators under 362.

I.e. reciprocals of perfect squares.

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Solution computer solution Comment 1 of 1
1/2 = 1/4 + 1/9 + 1/16 + 1/25 + 1/49 + 1/144 + 1/225 + 1/400 + 1/784 + 1/1225

The denominators are the squares of:

 2 3 4 5 7 12 15 20 28 35

as found by the below VB program and verified by using exact rational fractions in UBASIC.

The program  produces a total that prints out as exactly .5 but did not register as equal to .5 within the program, due to rounding, and thus necessitated the difference to be below the small threshhold of absolute value difference shown in the program, rather than testing if = 0.5.

The first portion of the program determined how far back from 1/36^2 would be necessary and found that the first term must be 1/2^2, otherwise the total would be impossible. It was only after this was run that the rest of the program was added, to find the actual result.


DefDbl A-Z
Dim crlf$, h(36), tot


Private Sub Form_Load()
 Form1.Visible = True
 
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 For rt = 36 To 1 Step -1
   ptot = tot
   tot = tot + 1 / (rt * rt)
   If tot >= 0.5 Then Text1.Text = Text1.Text & rt & Str(tot) & Str(ptot) & crlf & crlf: Exit For
 Next
 
 h(1) = 4: tot = 1 / 4
 addOn 2
 
 
 Text1.Text = Text1.Text & crlf & " done"
  
End Sub

Sub addOn(wh)
  DoEvents
  st = Int(Sqr(h(wh - 1)) + 0.5) + 1
  For i = st To 36
    h(wh) = i * i
    savetot = tot
    tot = tot + 1 / (i * i)
    If Abs(tot - 0.5) < 0.000000000001 Then
       Text1.Text = Text1.Text & tot & crlf & "   "
       For j = 1 To wh
         Text1.Text = Text1.Text & " + 1/" & h(j)
       Next
       Text1.Text = Text1.Text & crlf
       For j = 1 To wh
         Text1.Text = Text1.Text & Str(Sqr(h(j)))
       Next
       Text1.Text = Text1.Text & crlf
       
    End If
    If i < 36 And tot < 0.5001 Then
      addOn wh + 1
    End If
    tot = savetot
  Next
End Sub



2 .617538519845469 .367538519845469

0.5
    + 1/4 + 1/9 + 1/16 + 1/25 + 1/49 + 1/144 + 1/225 + 1/400 + 1/784 + 1/1225
 2 3 4 5 7 12 15 20 28 35

  Posted by Charlie on 2017-06-28 13:59:04
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