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Eight distances (Posted on 2014-12-10) Difficulty: 2 of 5
Within the top left-hand quarter of a square(its side an integer below 40 cm) I marked two points, each of which was distant an integral number of cm, not only from each side of the square but also from both the top left-hand corner and the bottom right-hand corner of the square.

What size was a side of this square?

Based on Enigma problem (New Scientist 2010)

No Solution Yet Submitted by Ady TZIDON    
Rating: 4.0000 (1 votes)

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Solution can't be sure | Comment 1 of 4
 For sz = 4 To 79
   For xc1 = 1 To sz / 2
   For yc1 = 1 To sz / 2
     d1sq = xc1 * xc1 + yc1 * yc1
     sr = Int(Sqr(d1sq) + 0.5)
     If sr * sr = d1sq Then
       d2sq = (sz - xc1) * (sz - xc1) + (sz - yc1) * (sz - yc1)
       sr = Int(Sqr(d2sq) + 0.5)
       If sr * sr = d2sq Then
         d1save = Sqr(d1sq): d2save = Sqr(d2sq)
   For xc2 = 1 To sz / 2
   For yc2 = 1 To sz / 2
     d1sq = xc2 * xc2 + yc2 * yc2
     sr = Int(Sqr(d1sq) + 0.5)
     If sr * sr = d1sq And (xc2 <> xc1 Or yc2 <> yc1) Then
       d2sq = (sz - xc2) * (sz - xc2) + (sz - yc2) * (sz - yc2)
       sr = Int(Sqr(d2sq) + 0.5)
       If sr * sr = d2sq Then
         
         Text1.Text = Text1.Text & sz & Str(xc1) & Str(yc1) & "   "
         Text1.Text = Text1.Text & d1save & Str(d2save) & "    "
         Text1.Text = Text1.Text & sz & Str(xc2) & Str(yc2) & "   "
         Text1.Text = Text1.Text & Sqr(d1sq) & Str(Sqr(d2sq)) & crlf
       End If
     End If
   Next
   Next
       End If
     End If
   Next
   Next
 Next sz

finds

24 3 4   5 29    24 4 3   5 29
24 4 3   5 29    24 3 4   5 29
33 5 12   13 35    33 12 5   13 35
33 12 5   13 35    33 5 12   13 35
36 8 15   17 35    36 15 8   17 35
36 15 8   17 35    36 8 15   17 35

----------------------------------

48 6 8   10 58    48 8 6   10 58
48 8 6   10 58    48 6 8   10 58
52 7 24   25 53    52 24 7   25 53
52 24 7   25 53    52 7 24   25 53
60 5 12   13 73    60 12 5   13 73
60 12 5   13 73    60 5 12   13 73
63 8 15   17 73    63 15 8   17 73
63 15 8   17 73    63 8 15   17 73
66 10 24   26 70    66 24 10   26 70
66 24 10   26 70    66 10 24   26 70
72 9 12   15 87    72 12 9   15 87
72 9 12   15 87    72 16 30   34 70
72 9 12   15 87    72 30 16   34 70
72 12 9   15 87    72 9 12   15 87
72 12 9   15 87    72 16 30   34 70
72 12 9   15 87    72 30 16   34 70
72 16 30   34 70    72 9 12   15 87
72 16 30   34 70    72 12 9   15 87
72 16 30   34 70    72 30 16   34 70
72 30 16   34 70    72 9 12   15 87
72 30 16   34 70    72 12 9   15 87
72 30 16   34 70    72 16 30   34 70
75 7 24   25 85    75 24 7   25 85
75 24 7   25 85    75 7 24   25 85
76 21 28   35 73    76 28 21   35 73
76 28 21   35 73    76 21 28   35 73
77 5 12   13 97    77 12 5   13 97
77 12 5   13 97    77 5 12   13 97

I've separated the results for squares with sides under 40 (as stated in the puzzle) from those with sides over 40 (as in the original Enigma puzzle).

But even below 40 there are more than one solution:

A 24x24 square could have points at (3,4) and (4,3) with distances from the mentioned corners of 5 and 29 cm.

A 33x33 could have points (5,12) and (12,5) for distances of 13 and 35.

A 36x36 could have points (8,15) and (15,8) for distances of 17 and 35.

So I can't be sure which one was actually constructed.

The original Enigma had asked for four such points in a square under 80x80, with the unique answer being 72x72 with points at (9,12), (12,9), (16,30) and (30,16).

  Posted by Charlie on 2014-12-10 10:12:45
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