Magic Squares with Lucas and Fibonacci Numbers

Question

I am quite curious about can we construct magic squares using only Lucas and Fibonacci numbers(of course not repeating them? If yes, how can we construct them?
And if not , what is the proof?

Answer 1

Theorem: It is not possible to make a nontrivial (1x1 magic squares always work) magic square out of purely Fibonacci numbers or purely Lucas numbers.

Proof: Assume for the sake of contradiction that we have a magic square $M$ with $n^2$ different Fibonacci numbers as elements, and $n\geq2$. Let $F_k$ be the largest entry in the magic square, and let $R$ be the row which includes $F_k$.

First suppose either $F_k$ or $F_{k-1}$ appears in every row. Then $M$ can only be a 2x2 square. This leads to a contradiction because there exist no nontrivial 2x2 magic squares (trivial squares here mean ones consisting of all the same element). We can see this by noting the sum of any two elements in a 2x2 magic square must equal the sum of any other two elements in the square, which implies all entries are the same.

Suppose now there exists a row $H$ that neither includes $F_k$ nor $F_{k-1}$. Then

$$\sum\limits_{x\in H}x\leq\sum\limits_{n=0}^{k-2}F_n=F_k-1<F_k<\sum\limits_{x\in R}x.$$

Hence the sum of elements in row $H$ is strictly less than the sum of elements in row $R$, so $M$ cannot be a magic square.

This completes the proof.

The case of Lucas numbers is completely analogous since as with Fibonacci numbers, we have $$\sum\limits_{n=0}^{k-2}L_n=L_k-1,$$ which can be proven easily by induction.

In essence, both the Fibonacci numbers and the Lucas numbers are spaced too far apart from each other to form a magic square.

Answer 2

You can't set up a 4x4 magic square, but you can set it up using the pattern for an odd-numbered magic square. I'll demonstrate with a 3x3:

Pick any 9 repeating Fibonacci numbers, I like 2, 3, 5, 8, 13, 21, 34, 55, 89 (1 gets a little messy).

Arrange them as you would any sequence of magic numbers

5 13 34

8 89 3

55 2 21

Now, instead of adding the rows and columns, multiply the numbers. You get:

Rows

5*13*34=2210

8*89*3=2136

55*2*21=2310

and

Columns

5*8*34=2200

13*89*2=2314

34*3*21=2142

Now take the totals we have for the columns and rows and add them

2200+2314+2142=6656

2210+2136+2310=6656

6656=6656

Boom, there ya go. Fibonacci magic square, or the closest thing you're going to get to it.