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 Only one basic solution (Posted on 2014-07-02)
A,B & C are 7-digit numbers, each being a permutation of 1234567.

If written in three rows, one below the other, the resulting column-wise sums
(i.e ai+bi+ci) will be square numbers.

Given A=2461357, find B and C.
Rem: Since B and C are interchangeable - there will be only one basic solution.

 See The Solution Submitted by Ady TZIDON No Rating

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 computer solutions Comment 2 of 2 |
DefDbl A-Z
Dim crlf\$, grid(3, 7), used(2 To 3, 9), solct

Text1.Text = "": Text2.Text = ""
crlf\$ = Chr(13) + Chr(10)
Form1.Visible = True
DoEvents

For i = 1 To 7
grid(1, i) = Val(Mid("2461357", i, 1))
Next

fillcol 1

Text1.Text = Text1.Text & solct

End Sub

Sub fillcol(col)
For b = 1 To 7
For sr = 2 To 5
sq = sr * sr
a = grid(1, col)
c = sq - b - a
If c < 8 And c > 0 And (c >= b Or col > 1) Then
If used(2, b) = 0 And used(3, c) = 0 Then
used(2, b) = 1: used(3, c) = 1
grid(2, col) = b: grid(3, col) = c
If col = 7 Then
row2 = 0
For i = 1 To 7
row2 = 10 * row2 + grid(2, i)
Next i
row3 = 0
For i = 1 To 7
row3 = 10 * row3 + grid(3, i)
Next i
If row3 >= row2 Then
Text1.Text = Text1.Text & "2461357" & crlf
For rw = 2 To 3
For cl = 1 To 7
Text1.Text = Text1.Text & grid(rw, cl)
Next: Text1.Text = Text1.Text & crlf
Next: Text1.Text = Text1.Text & crlf
DoEvents
solct = solct + 1
End If
Else
fillcol col + 1
End If
used(2, b) = 0: used(3, c) = 0
End If
End If
Next sr
Next b
End Sub

We have 67 basic (B <= C) solutions:

2461357
1234567
6374152

2461357
1236574
6372145

2461357
1237465
6371254

2461357
1254637
6354712

2461357
1263547
6345172

2461357
1325674
6213745

2461357
1357246
6251473

2461357
1374562
6234157

2461357
1435267
6173452

2461357
1456372
6152347

2461357
1473625
6135724

2461357
1523476
6715243

2461357
1524367
6714352

2461357
1526374
6712345

2461357
1567324
6741325

2461357
1576234
6732415

2461357
1726345
6512374

2461357
1734526
6574123

2461357
1765432
6543217

2461357
2137456
5471263

2461357
2145637
5463712

2461357
2176543
5432176

2461357
2341576
5267143

2461357
2345671
5263741

2461357
2347615
5261734

2461357
2365741
5243671

2461357
2374651
5234761

2461357
2415673
5123746

2461357
2436715
5172634

2461357
2461357
5147362

2461357
2514736
5724613

2461357
2546371
5762341

2461357
2567413
5741236

2461357
2617345
5621374

2461357
2634517
5674132

2461357
2635471
5673241

2461357
2637415
5671234

2461357
2671435
5637214

2461357
3217654
4321765

2461357
3241567
4367152

2461357
3256741
4352671

2461357
3415762
4123657

2461357
3451726
4157623

2461357
3452617
4156732

2461357
3456712
4152637

2461357
3476152
4132567

2461357
3526714
4712635

2461357
3547126
4761523

2461357
3572461
4736251

2461357
3625147
4613572

2461357
3657412
4651237

2461357
3671524
4637125

2461357
3712546
4526173

2461357
3721456
4517263

2461357
3741526
4567123

2461357
3745621
4563721

2461357
3746512
4562137

2461357
7123456
7415263

2461357
7125463
7413256

2461357
7126354
7412365

2461357
7143526
7465123

2461357
7152436
7456213

2461357
7214563
7324156

2461357
7246135
7362514

2461357
7263451
7345261

2461357
7615234
7623415

2461357
7654321
7654321

This last solution has B equal to C, and the digits are in descending order.

If we require that no two digits in the same column be equal, we're reduced to six basic solutions:

2461357
1236574
6372145

2461357
1325674
6213745

2461357
1523476
6715243

2461357
1576234
6732415

2461357
3526714
4712635

2461357
3712546
4526173

 Posted by Charlie on 2014-07-02 12:33:17

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