Magic Square Trivia Quiz

Answer: M(n) = (n^3 + n)/2

The magic constants for magic squares of order n = 3, 4, 5 and 6 are 15, 34, 65 and 111 respectively.

Answer: True

A magic square with an order of 1 is made up of a single cell with a single number, namely 1. This is known as the trivial case. The smallest nontrivial case is that with an order of 3, as shown below:

4 3 8
9 5 1
2 7 6
The magic constant, M is 15. (for n = 3)

The magic square with an order of 4 is:

16 05 09 04
02 11 07 14
03 10 06 15
13 08 12 01
The magic number is 34. (for n = 4)

Answer: 1

For n = 4, there are 880 possible solutions. On the other hand, there are 275305224 solutions for the case of n = 5. It is estimated that for n = 6, there are as many as 1.8 x 10^19 solutions.

Answer: 25

One of the magic squares of order 7 is shown below:

04 29 12 37 20 45 28
35 11 36 19 44 27 03
10 42 18 43 26 02 34
41 17 49 25 01 33 09
16 48 24 07 32 08 40
47 23 06 31 14 39 15
22 05 30 13 38 21 46

There are many possible solutions for the arrangements of numbers 1 to 49 in the magic square above. However, all of those solutions have the same middle value, namely 25.

The formula for the middle value is given by [(n^2)/2] + 0.5, where n is the order of the magic square. The middle value for the magic square of order n is also the middle value for the list of consecutive numbers from 1 until n^2.

Answer: True

We will use the fundamental move of "diagonally up and right", although "diagonally up and left might work, too.

Let us try to construct our own magic square. First of all, we fill in the number 1 in the middle of the first row.

X 1 X
X X X
X X X

Our next move will leave the square, so we need to "imagine" there are other identical magic squares that surround our magic square.

X 1 X
X X X
X X X
_____

X 1 X
X X X
X X X

Hence, we fill in the number 2 as below:

X 1 X
X X X
X X 2

Moving diagonally up and right again, we fill in the number 3 as below

X 1 X
3 X X
X X 2


Moving diagonally up and right again, we notice that the position that we are supposeD to fill with number 4 is already occupied with number 1. Whenever this happens, we fill in our next number at the bottom of our previous number. We will obtain

X 1 X
3 X X
4 X 2

Repeating the steps, we will get
X 1 X
3 5 X
4 X 2

Then
X 1 6
3 5 X
4 X 2

Next
X 1 6
3 5 7
4 X 2

Then
8 1 6
3 5 7
4 X 2

Eventually, we will get
8 1 6
3 5 7
4 9 2

That's it; we have constructed our own magic square with an order of 3. You can try to do another one with different order, by using the same step and the fundamental movement.

Answer: True

The smallest order of a bimagic square is 8.

Answer: Yes

There exist other trimagic squares with different orders, such as 12, 32, 64, 81 and 128.

Answer: Heterosquare

An example of heterosquare with an order of 3 is:

01 02 03 | 6
08 09 04 | 21
07 06 05 | 18
__ __ __
16 17 12

Besides, the sums of the 2 diagonals are 15 and 19 respectively.

Answer: Latin square

Another example of Latin square with an order of 4 is:

01 02 03 04
02 01 04 03
03 04 01 02
04 03 02 01

Leonhard Euler used Latin characters to fill in the square. That's how we get the name Latin square.

Answer: 14

Imagine a 3 x 3 x 3 Rubik's cube, where each small cube represents a number. The magic cube with an order of 3 takes the following form:
First layer:
08 24 10
12 07 23
22 11 09

Second layer:
15 01 26
25 14 03
02 27 13

Third layer:
19 17 06
05 21 16
18 04 20

Notice that the middle value is 14. Moreover, the numbers on each row, column, and space diagonal add up to 42.