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Thread: Magic Square  Medium Difficulty

20030727, 03:17 #1
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Magic Square  Medium Difficulty
In the 7 X 7 square below, you are to insert the integers from 30 through 54 (in the white squares, only) so that each row and column adds to 150. And the two main diagonals add to 300.
 Ricky

20030801, 08:40 #2
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Re: Magic Square  Medium Difficulty
Here is one solution.

20030801, 10:54 #3
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Re: Magic Square  Medium Difficulty
In case people are interested in how I found the solution:
<span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>The problem can be broken down to two magic squares: the numbers in the first, third, fifth and seventh row and column form a 4 x 4 magic square, and those in the second, fourth and sixth row and column form a 3 x 3 magic square. The 3 x 3 square must use the highest numbers 46 through 54, otherwise you'll never get the required row and column sums of 150. It is not difficult to construct such a square. This leaves the numbers 30 through 45 for the 4 x 4 square. Again, it is not difficult to construct such a square. If you subtract 29 from each number you get a "classic" 4 x 4 magic square filled with the numbers 1 through 16, with row and column sums of 34. Mathematicians have algorithms to construct such squares methodically, but you can easily create one by trial and error. Then add 29 to each number and you're done.</font color=yellow></span hi>

20030801, 17:48 #4
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Re: Magic Square  Medium Difficulty
I worked on this one for a long while and finally solved it using Excel. No method other than trying every possible combination. Never occurred to me to break it into separate magic squares.
This puzzle as well as the triangle puzzle came from a math puzzles book. I just took a look at the solution for the checkerboard magic square and it also explains breaking the square into a 4x4 and a 3x3. So, you nailed it. <img src=/S/cheers.gif border=0 alt=cheers width=30 height=16>
On the triangle, I found that solving the three inner triangles resulted in the trapezoids working themselves out by accident. Got lucky, I guess. Ricky

20030801, 21:02 #5
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Re: Magic Square  Medium Difficulty
>> On the triangle, I found that solving the three inner triangles resulted in the trapezoids working themselves out by accident.
It's not by accident. The sum of the digits 1, 2, ..., 9 is 45, so if the sum of the digits in a triangle is 23, the sum of the digits in the corresponding trapezoid will automatically be 45  23 = 22 <img src=/S/grin.gif border=0 alt=grin width=15 height=15>