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4 numbers make 5 squares (Posted on 2016-06-30) Difficulty: 3 of 5
Find four whole numbers such that:

each pairwise sum is a perfect square and
the sum of all four is a perfect square.

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Solution computer solution and discussion | Comment 1 of 10
While this puzzle was in the queue there was some question as to whether non-integers would have any bearing.  Half-integers supply alternatives, as I've decided to show below.  I don't think any other non-integers would be applicable. But of course if you take a solution using half-integers and multiply by 4, you get a solution in whole numbers.

In addition to cases where all are the same number, there are many cases where there are three of one number and one of another, such as 4, 32, 32, 32.  I've marked with asterisks cases where there are three different numbers, with the middle two (when arranged ascending) the same, such as -151, 200, 200, 376 to take an integral example.  The only example in the range looked at where all the numbers are positive is the non-integral 328.5, 760.5, 760.5, 2375.5. But if you multiply each of these by 4, that should work too. Indeed 1314, 3042, 3042, 9502 does work, as well as 2569, 5000, 5000, 8456.

I didn't find any cases where all four numbers are different.

-2964.5 2964.5 2964.5 2964.5
-2888 2888 2888 2888
-2812.5 2812.5 2812.5 2812.5
-2738 2738 2738 2738
-2664.5 2664.5 2664.5 2664.5
-2592 2592 2592 2592
-2520.5 2520.5 2520.5 2520.5
-2450 2450 2450 2450
-2380.5 2380.5 2380.5 2380.5
-2312 2312 2312 2312
-2244.5 2244.5 2244.5 2244.5
-2178 2178 2178 2178
-2151 2592 2592 2592
-2112.5 2112.5 2112.5 2112.5
-2048 2048 2048 2048
-1984.5 1984.5 1984.5 1984.5
-1984.5 1984.5 1984.5 2240.5
-1922 1922 1922 1922
-1874 2450 2450 2450
-1860.5 1860.5 1860.5 1860.5
-1800 1800 1800 1800
-1800 1800 1800 1921
-1800 1800 1800 2425
-1800 1800 1800 2824
-1740.5 1740.5 1740.5 1740.5
-1728.5 1984.5 1984.5 1984.5
-1692 2592 2592 2592
-1682 1682 1682 1682
-1679 1800 1800 1800
-1668.5 2964.5 2964.5 2964.5
-1624.5 1624.5 1624.5 1624.5
-1568 1568 1568 1568
-1568 1568 1568 2657
-1544 1800 1800 2169        *******
-1512.5 1512.5 1512.5 1512.5
-1458 1458 1458 1458
-1404.5 1404.5 1404.5 1404.5
-1352 1352 1352 1352
-1352 1352 1352 2873
-1312 1568 1568 2401        *******
-1300.5 1300.5 1300.5 1300.5
-1250 1250 1250 1250
-1212.5 2812.5 2812.5 2812.5
-1200.5 1200.5 1200.5 1200.5
-1175 1800 1800 1800
-1152 1152 1152 1152
-1152 1152 1152 1348
-1152 1152 1152 1552
-1152 1152 1152 2448
-1104.5 1104.5 1104.5 1104.5
-1096 1352 1352 2617        *******
-1058 1058 1058 1058
-1012.5 1012.5 1012.5 1012.5
-1012.5 1012.5 1012.5 1588.5
-1012.5 1012.5 1012.5 1796.5
-968 968 968 968
-968 968 968 2057
-956 1152 1152 1152
-943 1568 1568 2032        *******
-924.5 924.5 924.5 924.5
-882 882 882 882
-882 882 882 2482
-840.5 840.5 840.5 840.5
-800 800 800 800
-800 800 800 881
-800 800 800 1700
-800 800 800 2564
-776 1800 1800 1800
-760.5 760.5 760.5 760.5
-752 1152 1152 1152
-727 1352 1352 2248        *******
-722 722 722 722
-719 800 800 800
-684.5 684.5 684.5 684.5
-648 648 648 648
-648 648 648 873
-648 648 648 1377
-648 648 648 2952
-612.5 612.5 612.5 612.5
-612.5 612.5 612.5 756.5
-604 800 800 1504        *******
-578 578 578 578
-544.5 544.5 544.5 544.5
-544.5 544.5 544.5 2480.5
-512 512 512 512
-512 512 512 1088
-480.5 480.5 480.5 480.5
-479 1568 1568 1568
-468.5 612.5 612.5 612.5
-450 450 450 450
-450 450 450 706
-450 450 450 2050
-436.5 1012.5 1012.5 1012.5
-423 648 648 648
-420.5 420.5 420.5 420.5
-392 392 392 392
-392 392 392 833
-392 392 392 2417
-364.5 364.5 364.5 364.5
-364.5 364.5 364.5 1660.5
-338 338 338 338
-312.5 312.5 312.5 312.5
-288 288 288 288
-288 288 288 337
-288 288 288 388
-288 288 288 612
-288 288 288 1312
-288 288 288 2313
-264.5 264.5 264.5 264.5
-263 1352 1352 1784        *******
-254 450 450 1854        *******
-242 242 242 242
-239 288 288 288
-228.5 1012.5 1012.5 1012.5
-220.5 220.5 220.5 220.5
-220.5 220.5 220.5 620.5
-220.5 220.5 220.5 1004.5
-200 200 200 200
-200 200 200 425
-200 200 200 641
-200 200 200 2504
-194 450 450 450
-188 288 288 288
-180.5 180.5 180.5 180.5
-162 162 162 162
-162 162 162 738
-151 200 200 376        *******
-144.5 144.5 144.5 144.5
-135.5 760.5 760.5 2839.5        *******
-128 128 128 128
-128 128 128 272
-128 128 128 1028
-112.5 112.5 112.5 112.5
-112.5 112.5 112.5 176.5
-112.5 112.5 112.5 512.5
-112.5 112.5 112.5 1408.5
-98 98 98 98
-98 98 98 2402
-84.5 84.5 84.5 84.5
-72 72 72 72
-72 72 72 97
-72 72 72 153
-72 72 72 328
-72 72 72 1297
-63.5 112.5 112.5 463.5        *******
-60.5 60.5 60.5 60.5
-50 50 50 50
-50 50 50 626
-48.5 112.5 112.5 112.5
-47 72 72 72
-40.5 40.5 40.5 40.5
-40.5 40.5 40.5 184.5
-40.5 40.5 40.5 1640.5
-32 32 32 32
-32 32 32 68
-32 32 32 257
-24.5 24.5 24.5 24.5
-24.5 24.5 24.5 600.5
-18 18 18 18
-18 18 18 82
-12.5 12.5 12.5 12.5
-12.5 12.5 12.5 156.5
-8 8 8 8
-8 8 8 17
-4.5 4.5 4.5 4.5
-4.5 4.5 4.5 20.5
-2 2 2 2
-0.5 .5 .5 .5
0 1 0 0
1 8 8 8
4 32 32 32
4.5 4.5 4.5 11.5
9 72 72 72
11.5 4.5 4.5 4.5
12.5 12.5 12.5 131.5
16 128 128 128
18 18 18 46
24.5 24.5 24.5 551.5
25 200 200 200
32 32 32 193
36 288 288 288
40.5 40.5 40.5 103.5
40.5 40.5 40.5 1559.5
49 392 392 392
50 50 50 526
64 512 512 512
72 72 72 184
72 72 72 1153
81 648 648 648
98 98 98 2206
100 800 800 800
112.5 112.5 112.5 287.5
112.5 112.5 112.5 1183.5
121 968 968 968
128 128 128 772
144 1152 1152 1152
162 162 162 414
169 1352 1352 1352
179.5 220.5 220.5 220.5
196 1568 1568 1568
200 200 200 241
200 200 200 2104
220.5 220.5 220.5 563.5
225 1800 1800 1800
241 200 200 200
256 2048 2048 2048
288 288 288 736
288 288 288 1737
289 2312 2312 2312
324 2592 2592 2592
328.5 760.5 760.5 2375.5        *******
361 2888 2888 2888
364.5 364.5 364.5 931.5
392 392 392 1633
450 450 450 1150
544.5 544.5 544.5 1391.5
544.5 544.5 544.5 2591.5
648 648 648 1656
718 882 882 882
760.5 760.5 760.5 1943.5
791.5 1512.5 1512.5 1512.5
800 800 800 964
882 882 882 2254
1012.5 1012.5 1012.5 2587.5
1152 1152 1152 1873
1152 1152 1152 2944
1615.5 1984.5 1984.5 1984.5
1633 2592 2592 2592
1800 1800 1800 2169

DefDbl A-Z
Dim crlf$


Private Sub Form_Load()
 Form1.Visible = True
 
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 lim = 3000
 lim2 = lim * lim
 
 For a = -lim To lim Step 0.5
  If a > 0 Then lval = Int(Sqr(2 * a)) Else lval = 0
  For sr1 = lval To Sqr(a + lim)
     b = sr1 * sr1 - a
     If b > 0 Then
     For sr2 = Int(Sqr(2 * b)) To Sqr(b + lim)
       c = sr2 * sr2 - b
       If a + c >= 0 Then
       sr = Int(Sqr(a + c) + 0.5)
       If sr * sr = a + c Then
         For sr3 = Int(Sqr(2 * c)) To Sqr(c + lim)
           d = sr3 * sr3 - c
            If d + a >= 0 Then
              sr = Int(Sqr(a + d) + 0.5)
              If sr * sr = a + d Then
                sr = Int(Sqr(b + d) + 0.5)
                If sr * sr = b + d Then
                  sr = Int(Sqr(a + b + b + d) + 0.5)
                  If sr * sr = a + b + c + d Then
                     Text1.Text = Text1.Text & a & Str(b) & Str(c) & Str(d)
                     If Abs(a) <> b And c <> d Then Text1.Text = Text1.Text & "        *******"
                     Text1.Text = Text1.Text & crlf
                  End If
                End If
              End If
            End If
         Next
       End If
       End If
     Next sr2
     End If
  Next sr1
 Next
 
 
 Text1.Text = Text1.Text & crlf & " done"
  
End Sub


  Posted by Charlie on 2016-06-30 11:20:00
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