perplexus.info :: Just Math : Only one basic solution

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 a_i+b_i+c_i) 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

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computer solutions

Comment 2 of 2 |

DefDbl A-Z

Dim crlf$, grid(3, 7), used(2 To 3, 9), solct

Private Sub Form_Load()

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))

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

DoEvents

solct = solct + 1

End If

Else

fillcol col + 1

End If

used(2, b) = 0: used(3, c) = 0

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

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

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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