When I was a boy, my grandfather used to give me puzzles for me to solve. The following puzzle is one of them.

A magic square is an arrangement of numbers from 1 to n2 in an n2 matrix in which each number should only occur once. Every row, column, and diagonal of a magic square have the same sum called the magic sum. The most popular magic square is the 3 × 3 magic square, also known as the Lo Shu Square:

It contains the numbers 1 to 9 and each row, column and diagonal of this magic square has a magic sum of 15.

While there are systematic methods for solving magic squares, you don’t need to know anything about them to solve this particular magic square as it can be solved by trial and error.

Now, here’s the puzzle my grandfather challenged me to solve:

When solving a regular 3 × 3 magic square, you would find that the number 8 must be in one of the corners of the magic square. The challenge is this: Can you create a magic square wherein the number 8 is on the top center (same confitions for constructing a 3 × 3 magic square apply) as shown:

Solution

Before you tell me that this is not an acceptable solution, remember that this solution has met all the conditions of the puzzle. Even if there are fractions, it still complied with the conditions that only numbers from 1 through 9 should be used and each number should only be used once. If I said that fractions are not allowed, then this puzzle would most likely be unsolvable. The solution is not obvious to most people which made this puzzle difficult.

When my grandfather showed this puzzle to me, I immediately said that it’s impossible to solve it. I know how to construct magic squares so I was sure that his puzzle has no valid solution. He just laughed and encouraged me to try harder.

Days, then weeks had gone by and I got nowhere in answering it. Then I began to pester him to tell me the answer. However, my grandfather would never spoil his puzzles so he just ignored my requests.

But as weeks became months, I set aside the puzzle at some point and ceased to think about it as I was distracted by more puzzles from my grandfather, books, several hobbies, and other diversions. A few more months later, my grandfather passed away.

I felt so dismal about it since we were very close and it was the first time I experienced how it felt to lose a loved one. During that time, I suddenly remember this particular unsolved puzzle and tried again. It took me more than a week to solve it. I’m not sure if this is the solution my grandfather wanted, but I believe that this is it.

Solving puzzles for my grandfather was one of my most cherished childhood memories. Seeing his joyful smile every time I solved a puzzle was enough to make me happy. No matter where he is now, I hope that he would somehow know that I finally solved this puzzle.

I’m very impressed that you solved the puzzle. I never could have done it!

Take care —

Neil S.

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I’m not a puzzle person myself, but your description of the puzzle and its connection to your relationship with your grandfather felt particularly moving to me. Congratulations on solving it, and I’m pretty sure your grandfather would be proud of your perseverance.

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Very touching story Edmark. It only makes this puzzle more meaningful and the fact that you eventually solved it, I’m sure your grandfather is smiling (wherever he is).

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Invert the puzzle so that the top row read 8 – 1 – 6. Then add 7 to each number. this will give you the top row as 15 – 8 – 13. Continue for the two remaining rows. The magic number is 36.

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Yes, it can be done that way. However, the condition of the puzzle required me to use only numbers from 1 through 9.

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Sorry. Missed that rule.

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I wonder how fractions feel when they are automatically not considered numbers. Knowing them, they are probably rational about it.

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They have mixed feelings about it I suppose…

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Wonderful tribute to you grandfather, thank you for sharing. 😊

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You’re welcome.

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I used to give the Lo Shu Square to visiting so-called normal students (visiting our school for the multiply handicapped) to solve; they enjoyed it very much! One visiting math pro later gave me a two or three-page mathematical-solution (which was far beyond trial and error). 🙂

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There are complicated methods out there, especially those for solving a 4n + 2 order magic squares (6×6, 10×10, etc.) but the methods for solving odd order magic squares are pretty easy.

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I used to have these exercises on my Math classes! Flash back!

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Congrats on solving it 🙂 And thank you for sharing the story behind it. Makes us realize that there are so many ways we feel the connection with our loved ones..

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Wow wonderful memory. Ty for sharing. I think I may just carry that along with my son.

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Interesting !!

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Touching and brilliant at the same time!

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Well done, 8mark! I believe that your grandpa knew then that you had solved the puzzle. And it’s very touching to know this beautiful memory with your grandpa is still vivid after all these years.

（*’∀’人）

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How sweet!

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💖😊

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The memory of your grandfather makes this puzzle special, plus the uncertainty of the unorthodox solution you are unable to validate and the source of your fascination with mathematics and puzzles.

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brilliant!

Peace and Equality and Love and Respect and Justice for All ✌ ✊ 👍

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Thanks for letting us know. It is in fact very interesting.

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Thanks and you are welcome.

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This is a very touching story, thank you for sharing this memory! Smart move with the answer!!

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Thanks for reading.

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Your Lo Shu puzzles are so very interesting, thanks for “magic square” article, I hadn’t seem them before….

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Glad you liked it.

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