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 One triplet out (Posted on 2015-06-16)
Given set of fifteen integers (1 to 15) .

Erase 3 numbers so that remaining integers can be arranged as a 3 by 4 array in which the sum of the numbers in each row is a certain Sr and in each of the columns a certain Sc.

Present the triplet you have chosen and one of the possible arrangements.

D4. Bonus question:
How many distinct solutions are there?

 No Solution Yet Submitted by Ady TZIDON No Rating

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 computer solution | Comment 2 of 3 |
DefDbl A-Z
Dim crlf\$, used(15), grid(3, 4), row1tot, col1tot, solct

Form1.Visible = True

Text1.Text = ""
crlf = Chr\$(13) + Chr\$(10)

Open "c:VB5 projects loobleone triplet out.txt" For Output As #2

Close

Text1.Text = Text1.Text & solct & crlf & " done"

End Sub

DoEvents
For newnum = 1 To 15
If used(newnum) = 0 Then
If row = 1 And col = 4 Then
Text2.Text = grid(1, 1) & Str(grid(1, 2)) & Str(grid(1, 3)) & Str(newnum)
End If
good = 1
If col = 4 Then
tot = grid(row, 1) + grid(row, 2) + grid(row, 3) + newnum
If row > 1 Then If tot <> row1tot Then good = 0
If row = 1 Then row1tot = tot
End If
If row = 3 Then
tot = grid(1, col) + grid(2, col) + newnum
If col > 1 Then If tot <> col1tot Then good = 0
If col = 1 Then col1tot = tot
End If
If row = 1 And col > 1 And newnum < grid(row, col - 1) Then good = 0
If col = 1 And row > 1 And newnum < grid(row - 1, col) Then good = 0
If good Then
grid(row, col) = newnum
used(newnum) = 1
If row = 3 And col = 4 Then
For r = 1 To 3
For c = 1 To 4
Print #2, Mid("123456789ABCDEF", grid(r, c), 1);
Next
Print #2, "    "; row1tot; col1tot
Next
Print #2,
solct = solct + 1
Else
c = col + 1
If c > 4 Then c = 1: r = row + 1 Else r = row
End If
used(newnum) = 0
End If
End If
Next
End Sub

finds there are 108 distinct solutions in which the columns are in order of ascending top numbers and the rows are in ascending order of leftmost number. The rows can be arranged in 3! = 6 ways, and the columns in 4! = 24 ways, so the total distinct solutions is 108 * 6 * 24 = 15,552. So you can see why it saves computation time to find only the 108.

While the arrays are given in hex below, for conciseness, the totals are of course in decimal:

```12EF     32  24
B975     32  24
CD34     32  24
-----------------
12EF     32  24
BD35     32  24
C974     32  24
-----------------
13DF     32  24
9A67     32  24
EB52     32  24
-----------------
13DF     32  24
9B57     32  24
EA62     32  24
-----------------
13DF     32  24
B795     32  24
CE24     32  24
-----------------
13DF     32  24
BE25     32  24
C794     32  24
-----------------
8695     28  21
CB23     28  21
-----------------
9586     28  21
BC32     28  21
-----------------
14CF     32  24
97A6     32  24
ED23     32  24
-----------------
14CF     32  24
A697     32  24
DE32     32  24
-----------------
14CF     32  24
A976     32  24
DB53     32  24
-----------------
14CF     32  24
AB56     32  24
D973     32  24
-----------------
14DE     32  24
8B67     32  24
F953     32  24
-----------------
14DE     32  24
B597     32  24
CF23     32  24
-----------------
158E     28  21
76B4     28  21
DA23     28  21
-----------------
15BF     32  24
96A7     32  24
ED32     32  24
-----------------
15BF     32  24
9D37     32  24
E6A2     32  24
-----------------
15BF     32  24
A796     32  24
DC43     32  24
-----------------
15BF     32  24
AC46     32  24
D793     32  24
-----------------
16AF     32  24
95B7     32  24
ED32     32  24
-----------------
16AF     32  24
97C4     32  24
EB25     32  24
-----------------
16AF     32  24
9D37     32  24
E5B2     32  24
-----------------
16AF     32  24
BE52     32  24
C497     32  24
-----------------
16BE     32  24
8D47     32  24
F593     32  24
-----------------
179F     32  24
A5B6     32  24
DC43     32  24
-----------------
179F     32  24
A6C4     32  24
DB35     32  24
-----------------
179F     32  24
AC46     32  24
D5B3     32  24
-----------------
179F     32  24
B3D5     32  24
CE24     32  24
-----------------
179F     32  24
BD53     32  24
C4A6     32  24
-----------------
179F     32  24
BE25     32  24
C3D4     32  24
-----------------
17AE     32  24
B498     32  24
CD52     32  24
-----------------
17BD     32  24
85A9     32  24
FC32     32  24
-----------------
17BD     32  24
8C39     32  24
F5A2     32  24
-----------------
17BD     32  24
8CA2     32  24
F539     32  24
-----------------
17BD     32  24
8E46     32  24
F395     32  24
-----------------
17BD     32  24
9F35     32  24
E2A6     32  24
-----------------
8E51     28  21
B467     28  21
-----------------
23BC     28  21
6895     28  21
DA14     28  21
-----------------
23BC     28  21
9568     28  21
-----------------
23CF     32  24
97B5     32  24
DE14     32  24
-----------------
23CF     32  24
9A58     32  24
DB71     32  24
-----------------
23DE     32  24
79A6     32  24
FC14     32  24
-----------------
23DE     32  24
7A69     32  24
FB51     32  24
-----------------
23DE     32  24
7B59     32  24
FA61     32  24
-----------------
23DE     32  24
A679     32  24
CF41     32  24
-----------------
24CE     32  24
95B7     32  24
DF13     32  24
-----------------
24CE     32  24
9F17     32  24
D5B3     32  24
-----------------
25AF     32  24
9C38     32  24
D7B1     32  24
-----------------
25BE     32  24
76A9     32  24
FD31     32  24
-----------------
25BE     32  24
79C4     32  24
FA16     32  24
-----------------
25BE     32  24
7D39     32  24
F6A1     32  24
-----------------
25BE     32  24
94C7     32  24
DF13     32  24
-----------------
25BE     32  24
97A6     32  24
DC34     32  24
-----------------
25BE     32  24
9F17     32  24
D4C3     32  24
-----------------
25BE     32  24
A697     32  24
CD43     32  24
-----------------
25BE     32  24
AF61     32  24
C479     32  24
-----------------
25CD     32  24
89B4     32  24
EA17     32  24
-----------------
269F     32  24
87C5     32  24
EB34     32  24
-----------------
26AE     32  24
75B9     32  24
FD31     32  24
-----------------
26AE     32  24
7D39     32  24
F5B1     32  24
-----------------
26AE     32  24
7DB1     32  24
F539     32  24
-----------------
28AC     32  24
9F35     32  24
D1B7     32  24
-----------------
28BF     36  27
C5A9     36  27
DE63     36  27
-----------------
34BE     32  24
9F62     32  24
C578     32  24
-----------------
34CD     32  24
697A     32  24
FB51     32  24
-----------------
34CD     32  24
6B5A     32  24
F971     32  24
-----------------
34CD     32  24
75B9     32  24
EF12     32  24
-----------------
34CD     32  24
79A6     32  24
EB25     32  24
-----------------
34CD     32  24
7F19     32  24
E5B2     32  24
-----------------
34CD     32  24
A679     32  24
BE52     32  24
-----------------
359F     32  24
76B8     32  24
ED41     32  24
-----------------
359F     32  24
7D48     32  24
E6B1     32  24
-----------------
359F     32  24
7DB1     32  24
E648     32  24
-----------------
359F     32  24
A6E2     32  24
BD17     32  24
-----------------
359F     32  24
AC28     32  24
B7D1     32  24
-----------------
35BD     32  24
679A     32  24
FC41     32  24
-----------------
35BD     32  24
6AC4     32  24
F917     32  24
-----------------
35BD     32  24
6C4A     32  24
F791     32  24
-----------------
35BD     32  24
74C9     32  24
EF12     32  24
-----------------
35BD     32  24
7F19     32  24
E4C2     32  24
-----------------
35BD     32  24
9F71     32  24
C46A     32  24
-----------------
36CF     36  27
A8B7     36  27
ED45     36  27
-----------------
36CF     36  27
B7A8     36  27
DE54     36  27
-----------------
36DE     36  27
9AC5     36  27
FB28     36  27
-----------------
379D     32  24
65BA     32  24
FC41     32  24
-----------------
379D     32  24
6C4A     32  24
F5B1     32  24
-----------------
379D     32  24
A2E6     32  24
BF15     32  24
-----------------
379D     32  24
AF16     32  24
B2E5     32  24
-----------------
389C     32  24
AF25     32  24
B1D7     32  24
-----------------
45DE     36  27
8AB7     36  27
FC36     36  27
-----------------
45DE     36  27
B78A     36  27
CF63     36  27
-----------------
468E     32  24
95F3     32  24
BD17     32  24
-----------------
46AC     32  24
5BD3     32  24
F719     32  24
-----------------
46AC     32  24
7F91     32  24
D35B     32  24
-----------------
46AC     32  24
93D7     32  24
BF15     32  24
-----------------
46AC     32  24
9F17     32  24
B3D5     32  24
-----------------
478D     32  24
93F5     32  24
BE16     32  24
-----------------
479C     32  24
53DB     32  24
FE21     32  24
-----------------
479C     32  24
5BE2     32  24
F61A     32  24
-----------------
479C     32  24
5E2B     32  24
F3D1     32  24
-----------------
479C     32  24
6FA1     32  24
E25B     32  24
-----------------
489B     32  24
7EA1     32  24
D25C     32  24
-----------------
56AB     32  24
73D9     32  24
CF14     32  24
-----------------
56AB     32  24
7F19     32  24
C3D4     32  24
-----------------
579B     32  24
62EA     32  24
DF13     32  24
-----------------
579B     32  24
6F1A     32  24
D2E3     32  24
-----------------
589A     32  24
71DB     32  24
CF23     32  24
-----------------
59AC     36  27
8FB2     36  27
E36D     36  27
-----------------
```

Jer's solution arranged above is the:

`158E     28  21 76B4     28  21 DA23     28  21 `

with the columns rearranged and the bottom two rows reversed.

Edited on June 16, 2015, 3:08 pm
 Posted by Charlie on 2015-06-16 15:05:58

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