This number N can be expressed as the sum of 1, 2, 3, 4, 5, 6, 7, 8 distinct squares.
Find N and show the 8 possible partitions.
Rem: This fact was discovered by Crespi de Valldaura.
289 is the lowest such number. In each set below, the terms making up the eight ways are shown below the number.
289
289
64 225
64 81 144
4 16 100 169
4 9 16 64 196
1 4 9 25 81 169
1 9 16 25 36 81 121
1 4 9 16 25 49 64 121
625
625
49 576
81 144 400
4 9 36 576
1 4 36 100 484
1 4 9 25 225 361
1 4 9 16 81 225 289
1 4 9 16 25 49 121 400
676
676
100 576
36 64 576
1 9 225 441
1 9 16 25 625
1 4 9 25 196 441
1 4 9 16 36 81 529
1 4 9 16 25 36 144 441
841
841
400 441
9 256 576
1 64 100 676
1 4 16 36 784
1 4 9 25 361 441
1 4 9 16 81 289 441
1 4 9 16 25 169 256 361
900
900
324 576
16 100 784
1 9 49 841
1 9 25 81 784
1 4 9 36 121 729
1 4 9 16 25 169 676
1 4 9 16 25 100 169 576
1156
1156
256 900
256 324 576
1 25 169 961
1 9 16 169 961
1 4 9 25 441 676
1 4 9 16 36 361 729
1 4 9 16 25 64 196 841
1225
1225
441 784
1 324 900
1 4 64 1156
1 4 36 400 784
1 4 9 25 225 961
1 4 9 16 25 81 1089
1 4 9 16 25 256 289 625
1369
1369
144 1225
9 64 1296
1 16 196 1156
1 4 16 324 1024
1 4 9 49 81 1225
1 4 9 16 25 225 1089
1 4 9 16 25 64 289 961
1521
1521
225 1296
4 361 1156
4 9 64 1444
1 4 36 324 1156
1 4 9 225 441 841
1 4 9 16 25 625 841
1 4 9 16 25 49 121 1296
1681
1681
81 1600
16 144 1521
1 16 64 1600
1 4 36 196 1444
1 4 9 25 121 1521
1 4 9 16 49 81 1521
1 4 9 16 25 256 529 841
2025
2025
729 1296
16 784 1225
1 4 256 1764
1 4 324 400 1296
1 4 9 49 441 1521
1 4 9 16 25 121 1849
1 4 9 16 25 49 400 1521
2500
2500
196 2304
324 576 1600
1 25 625 1849
1 9 16 625 1849
1 4 9 36 49 2401
1 4 9 16 36 225 2209
1 4 9 16 25 64 1156 1225
DefDbl A-Z
Dim crlf$, sq, amt(8, 8), solCt, remain, termCt, sol(8), series(8)
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf$ = Chr(13) + Chr(10)
For sr = 5 To 50
sq = sr * sr
For i = 1 To 8
sol(i) = 0
Next
remain = sq: solCt = 0
test
If solCt = 8 Then
Text1.Text = Text1.Text & sq & crlf
For i = 1 To 8
For j = 1 To i
Text1.Text = Text1.Text & Str(amt(i, j))
Next
Text1.Text = Text1.Text & crlf
Next
Text1.Text = Text1.Text & crlf
DoEvents
End If
Next
Text1.Text = Text1.Text & " done"
End Sub
Sub test()
DoEvents
For trsr = Int(Sqr(series(termCt)) + 0.5) + 1 To Sqr(remain)
DoEvents
remain = remain - trsr * trsr
termCt = termCt + 1
series(termCt) = trsr * trsr
If remain > 0 Then
If termCt < 8 Then
test
End If
ElseIf remain = 0 Then
If sol(termCt) = 0 Then
sol(termCt) = 1
solCt = solCt + 1
For i = 1 To termCt
amt(termCt, i) = series(i)
Next
End If
Else
Exit For
End If
termCt = termCt - 1
remain = remain + trsr * trsr
Next
End Sub
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Posted by Charlie
on 2018-12-06 22:40:19 |
computer solution
