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1 To 15 in a 3x5 Array (Posted on 2004-12-09) Difficulty: 2 of 5

Place the numbers 1 through 15 in a 3 by 5 array so that each column has the same sum and each row has the same sum.

See The Solution Submitted by Erik O.    
Rating: 3.0000 (3 votes)

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Solution All the solutions--computer solution | Comment 5 of 11 |

There are 17 fundamental solutions, each of which can be trivially modified by 720 permutations of its rows and columns for a total of 12,240 variants.

Arranged as 3 rows x 5 columns, with the "1" placed at the upper left, and the first row in ascending sequence and the first column in ascending sequence, the 17 fundamental solutions are:

 1             3             10            11            15 
 9             8             12            6             5
 14            13            2             7             4
 1             3             10            11            15 
 9             8             12            7             4
 14            13            2             6             5
 1             4             9             11            15 
 10            6             12            5             7
 13            14            3             8             2
 1             5             9             11            14 
 8             12            13            3             4
 15            7             2             10            6
 1             5             9             11            14 
 10            4             12            6             8
 13            15            3             7             2
 1             5             10            11            13 
 9             4             12            7             8
 14            15            2             6             3
 1             5             10            11            13 
 9             15            2             6             8
 14            4             12            7             3
 1             6             8             10            15 
 9             11            3             12            5
 14            7             13            2             4
 1             6             8             10            15 
 9             11            13            2             5
 14            7             3             12            4
 1             6             8             10            15 
 11            4             13            5             7
 12            14            3             9             2
 1             6             8             10            15 
 11            14            3             5             7
 12            4             13            9             2
 1             6             9             10            14 
 8             13            4             12            3
 15            5             11            2             7
 1             7             8             9             15 
 10            11            2             12            5
 13            6             14            3             4
 1             7             8             9             15 
 11            4             14            5             6
 12            13            2             10            3
 1             7             8             10            14 
 11            15            3             5             6
 12            2             13            9             4
 1             7             9             10            13 
 8             3             11            12            6
 15            14            4             2             5
 1             7             9             10            13 
 8             14            11            2             5
 15            3             4             12            6

The program is:

DIM numb(15)
CLS
OPEN "1-15i3x5.txt" FOR OUTPUT AS #2

FOR b1 = 2 TO 9
numb(b1) = 1

FOR c1 = b1 + 1 TO 10
IF numb(c1) = 0 THEN
numb(c1) = 1

FOR d1 = c1 + 1 TO 11
IF numb(d1) = 0 THEN
numb(d1) = 1

e1 = 40 - 1 - b1 - c1 - d1
IF e1 >= 1 AND e1 <= 15 THEN
IF numb(e1) = 0 THEN
numb(e1) = 1

FOR a2 = 2 TO 11
IF numb(a2) = 0 THEN
numb(a2) = 1

FOR b2 = 2 TO (24 - b1 - 2)
IF b2 <= 15 THEN
IF numb(b2) = 0 THEN
numb(b2) = 1

FOR c2 = 2 TO (24 - c1 - 2)
IF c2 <= 15 THEN
IF numb(c2) = 0 THEN
numb(c2) = 1

FOR d2 = 2 TO (24 - d1 - 2)
IF d2 <= 15 THEN
IF numb(d2) = 0 THEN
numb(d2) = 1

e2 = 40 - a2 - b2 - c2 - d2
IF e2 <= 15 AND e2 >= 0 THEN
IF numb(e2) = 0 THEN
numb(e2) = 1

a3 = 24 - 1 - a2
b3 = 24 - b1 - b2
c3 = 24 - c1 - c2
d3 = 24 - d1 - d2
e3 = 24 - e1 - e2

IF 24 - 1 - a2 <= 15 AND 24 - b1 - b2 <= 15 AND 24 - c1 - c2 <= 15 AND 24 - d1 - d2 <= 15 AND 24 - e1 - e2 <= 15 THEN
IF 24 - 1 - a2 > 0 AND 24 - b1 - b2 > 0 AND 24 - c1 - c2 > 0 AND 24 - d1 - d2 > 0 AND 24 - e1 - e2 > 0 THEN
IF numb(24 - 1 - a2) = 0 AND numb(24 - b1 - b2) = 0 AND numb(24 - c1 - c2) = 0 AND numb(24 - d1 - d2) = 0 AND numb(24 - e1 - e2) = 0 THEN

IF a3 <> b3 AND a3 <> c3 AND a3 <> d3 AND a3 <> e3 AND b3 <> c3 AND b3 <> d3 AND b3 <> e3 AND c3 <> d3 AND c3 <> e3 AND d3 <> e3 THEN

PRINT 1, b1, c1, d1, e1
PRINT a2, b2, c2, d2, e2
PRINT 24 - a2 - 1, 24 - b2 - b1, 24 - c2 - c1, 24 - d2 - d1, 24 - e2 - e1
PRINT
PRINT #2, 1, b1, c1, d1, e1
PRINT #2, a2, b2, c2, d2, e2
PRINT #2, 24 - a2 - 1, 24 - b2 - b1, 24 - c2 - c1, 24 - d2 - d1, 24 - e2 - e1
PRINT #2,
ct = ct + 1

END IF

END IF
END IF
END IF

numb(e2) = 0
END IF
END IF

numb(d2) = 0
END IF
END IF
NEXT
    
numb(c2) = 0
END IF
END IF
NEXT

numb(b2) = 0
END IF
END IF
NEXT

numb(a2) = 0
END IF
NEXT

numb(e1) = 0
END IF
END IF

numb(d1) = 0
END IF
NEXT

numb(c1) = 0
END IF
NEXT

numb(b1) = 0
NEXT

PRINT ct
CLOSE

 


  Posted by Charlie on 2004-12-09 15:23:29
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