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Solutions to NPR Puzzle
for Magic Product Squares on 5040

The NPR weekly listener challenge given on 2005-10-09, as recorded by Kristy Fowler Coker, is as follows:

This week’s challenge is from Ed Pegg, Jr., who runs MathPuzzle.com. In a standard 4x4 Magic Square, you arrange the digits from 1 to 16 so that each row, column, & corner-to-corner diagonal totals 34. This, however, is a *multiplication* Magic Square. Arrange 16 numbers in a four-by-four square so that the PRODUCT of each row, column, & corner-to-corner diagonal is 5,040. Use any numbers you want, but they have to be whole numbers and you can’t repeat a number in the square. Hint: The number in the upper left corner is 42.

Below are 80 solutions. The first 78 are those that came from a BASIC program I wrote. Obviously I missed how the 42 was to be in the upper LEFT corner. These are solutions with the 42 in the upper RIGHT corner. Below those are two solutions provided by Martin Eiger. If you would like to subscribe to our free weekly synopsis of the NPR puzzle, send a blank email message to:
nprpuzzle-subscribe@igc.topica.com

1 8 15 42
21 30 4 2
24 3 14 5
10 7 6 12

1 8 15 42
30 21 2 4
14 5 24 3
12 6 7 10

1 10 12 42
21 24 5 2
30 3 14 4
8 7 6 15

1 10 12 42
24 21 2 5
14 4 30 3
15 6 7 8

1 12 10 42
14 30 4 3
24 2 21 5
15 7 6 8

1 12 10 42
30 14 3 4
21 5 24 2
8 6 7 15

1 15 8 42
14 24 5 3
30 2 21 4
12 7 6 10

1 15 8 42
24 14 3 5
21 4 30 2
10 6 7 12

2 4 15 42
21 30 8 1
24 3 7 10
5 14 6 12

2 4 15 42
30 21 1 8
7 10 24 3
12 6 14 5

2 5 12 42
21 24 10 1
30 3 7 8
4 14 6 15

2 5 12 42
24 21 1 10
7 8 30 3
15 6 14 4

2 12 5 42
7 30 8 3
24 1 21 10
15 14 6 4

2 12 5 42
30 7 3 8
21 10 24 1
4 6 14 15

2 15 4 42
7 24 10 3
30 1 21 8
12 14 6 5


2 15 4 42
24 7 3 10
21 8 30 1
5 6 14 12

3 4 10 42
14 30 12 1
24 2 7 15
5 21 6 8

3 4 10 42
30 14 1 12
7 15 24 2
8 6 21 5

3 5 8 42
7 12 2 30
24 6 35 1
10 14 9 4

3 5 8 42
14 24 15 1
30 2 7 12
4 21 6 10

3 5 8 42
24 14 1 15
7 12 30 2
10 6 21 4

3 8 5 42
7 30 12 2
24 1 14 15
10 21 6 4

3 8 5 42
24 35 6 1
7 2 12 30
10 9 14 4

3 8 5 42
30 7 2 12
14 15 24 1
4 6 21 10

3 10 4 42
7 24 15 2
30 1 14 12
8 21 6 5

3 10 4 42
24 7 2 15
14 12 30 1
5 6 21 8

4 1 30 42
9 35 2 8
14 6 12 5
10 24 7 3

4 1 30 42
15 21 2 8
14 10 12 3
6 24 7 5

4 2 15 42
21 30 1 8
6 12 14 5
10 7 24 3

4 2 15 42
30 21 8 1
14 5 6 12
3 24 7 10

4 3 10 42
14 30 1 12
6 8 21 5
15 7 24 2

4 3 10 42
30 14 12 1
21 5 6 8
2 24 7 15

4 10 3 42
6 21 8 5
14 1 30 12
15 24 7 2

4 10 3 42
21 6 5 8
30 12 14 1
2 7 24 15

4 15 2 42
6 14 12 5
21 1 30 8
10 24 7 3

4 15 2 42
14 6 5 12
30 8 21 1
3 7 24 10

4 30 1 42
14 12 6 5
9 2 35 8
10 7 24 3

4 30 1 42
14 12 10 3
15 2 21 8
6 7 24 5

5 2 12 42
21 24 1 10
6 15 14 4
8 7 30 3

5 2 12 42
24 21 10 1
14 4 6 15
3 30 7 8

5 3 8 42
7 12 2 30
24 10 21 1
6 14 15 4

5 3 8 42
14 24 1 15
6 10 21 4
12 7 30 2

5 3 8 42
24 14 15 1
21 4 6 10
2 30 7 12

5 8 3 42
6 21 10 4
14 1 24 15
12 30 7 2

5 8 3 42
21 6 4 10
24 15 14 1
2 7 30 12

5 8 3 42
24 21 10 1
7 2 12 30
6 15 14 4

5 12 2 42
6 14 15 4
21 1 24 10
8 30 7 3

5 12 2 42
14 6 4 15
24 10 21 1
3 7 30 8

8 1 15 42
21 30 2 4
6 12 7 10
5 14 24 3

8 1 15 42
30 21 4 2
7 10 6 12
3 24 14 5

8 3 5 42
7 30 2 12
6 4 21 10
15 14 24 1

8 3 5 42
30 7 12 2
21 10 6 4
1 24 14 15

8 5 3 42
6 21 4 10
7 2 30 12
15 24 14 1

8 5 3 42
21 6 10 4
30 12 7 2
1 14 24 15

8 15 1 42
6 7 12 10
21 2 30 4
5 24 14 3

8 15 1 42
7 6 10 12
30 4 21 2
3 14 24 5

10 1 12 42
21 24 2 5
6 15 7 8
4 14 30 3

10 1 12 42
24 21 5 2
7 8 6 15
3 30 14 4

10 3 4 42
24 7 15 2
21 8 6 5
1 30 14 12

10 4 3 42
6 21 5 8
7 2 24 15
12 30 14 1

10 4 3 42
21 6 8 5
24 15 7 2
1 14 30 12

10 12 1 42
6 7 15 8
21 2 24 5
4 30 14 3

12 1 10 42
14 30 3 4
6 8 7 15
5 21 24 2

12 1 10 42
30 14 4 3
7 15 6 8
2 24 21 5

12 2 5 42
7 30 3 8
6 4 14 15
10 21 24 1

12 2 5 42
30 7 8 3
14 15 6 4
1 24 21 10

12 5 2 42
6 14 4 15
7 3 30 8
10 24 21 1

12 5 2 42
14 6 15 4
30 8 7 3
1 21 24 10

12 10 1 42
6 7 8 15
14 3 30 4
5 24 21 2

12 10 1 42
7 6 15 8
30 4 14 3
2 21 24 5

15 1 8 42
14 24 3 5
6 10 7 12
4 21 30 2

15 1 8 42
24 14 5 3
7 12 6 10
2 30 21 4

15 2 4 42
7 24 3 10
6 5 14 12
8 21 30 1

15 2 4 42
24 7 10 3
14 12 6 5
1 30 21 8

15 4 2 42
6 14 5 12
7 3 24 10
8 30 21 1

15 4 2 42
14 6 12 5
24 10 7 3
1 21 30 8

15 8 1 42
6 7 10 12
14 3 24 5
4 30 21 2

15 8 1 42
7 6 12 10
24 5 14 3
2 21 30 4

* ~ * ~ *

From: Martin Eiger
Subject: NPR magic product square
Date sent: Mon, 17 Oct 2005 12:11:41 -0400

The 78 solutions to the 5040 multiplication magic square that you've posted to your website seem to overlook two additional answers. Using your notation of putting the 42 in the upper-right corner ...

10 12 1 42
7 6 8 15
24 5 21 2
3 14 30 4

10 3 4 42
7 24 2 15
6 5 21 8
12 14 30 1

* ~ * ~ *

Click here to send me an email.

©2005 Richard Renner. Permission granted to Ed Pegg, Jr., NPR and Will Shortz to use these solutions. All Other Rights Reserved.

Updated 2005-10-23