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You will need to get assistance from your school if you are having problems entering the answers into your online assignment. Phone support is available Monday-Friday, 9:00AM-10:00PM ET. You may speak with a member of our customer support team by calling 1-800-876-1799. Boggle is a word game where players race to find words hidden in a grid of letters. Enter a Boggle board and see a list of all the words that can be found! Choose a board size - 3x3, 4x4 or 5x5 - and the minimum number of letters each word must have, and get a complete listing of all of the solutions along with point totals and other statistics.

Magic Square Generator

Magic Square Solver

Tool to generate magic squares size N, kind of matrices composed of distinct integers set such as the sum of any line or column are equal.

Answers to Questions

How to create a magic square of odd order?

Formula: Set number 1 on the left of the median line, the other numbers are written following the rule: if the cell is empty, in the cell in the bottom right of the previous one, else directly to the left of the occupied cell. When the cell does not exist (after last column or last row), go to the other side of the square/matrix.

What are the minimal possible sums (magic values)?

The constant values $ M $ of the sums of the magic squares have a minimum value (for non-zero integer positive values).

$$ M = n (n ^ 2 + 1) / 2 $$

For a size 3x3, the minimum constant is 15, for 4x4 it is 34, for 5x5 it is 65, 6x6 it is 111, then 175, 260, ...

Any lower sum will force the use of either negative numbers or fractions (not whole numbers) to solve the magic square.

What is the Franklin Square?

Franklin's square is a panmagic square with a magic constant of 260.

Example:

52	61	4	13	20	29	36	45
14	3	62	51	46	35	30	19
53	60	5	12	21	28	37	44
11	6	59	54	43	38	27	22
55	58	7	10	23	26	39	42
9	8	57	56	41	40	25	24
50	63	2	15	18	31	34	47
16	1	64	49	48	33	32	17

What is the Lo-Shu Magic Square?

This is a 3x3

Boggle Solver 5 5

magic square used in Feng Shui which is represented as well

4 Wealth	9 Fame	2 Relationship
3 Family	5 Health	7 Children
8 Wiseness	1 Career	6 Help/Friends

What is the KaldorMagic Square?

Kaldor's magic square is a square used in economics, which has nothing to do with digits or numbers of mathematics but rather with concepts from economic policy.

Is there a Magic Square in 3D (Magic Cube)?

Yes, there are magic cubes, their magic value is $$ M = n(n^3+1)/2 $$ (which may or may not have magic diagonals)

Example:

9 17 6 5 21 16 18 4 20

15 1 26 25 14 3 2 27 13

8 24 10 12 7 23 22 11 9

Source code

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Online 5x5 Solver Calculator

Questions / Comments

Methods explanation

Laplace Expansion Theorem

The Laplacian development theorem provides a method for calculating the determinant, in which the determinant is developed after a row or column. The dimension is reduced and can be reduced further step by step up to a scalar.

$\mathrm{det\; A}=\sum _{i=1}^n{-1}^{i+j}\cdot a_{ij}\det A_{ij}\text{( Expansion on the j-th column )}$

$\mathrm{det\; A}=\sum _{j=1}^n{-1}^{i+j}\cdot a_{ij}\det A_{ij}\text{( Expansion on the i-th row )}$

where A_ij, the sub-matrix of A, which arises when the i-th row and the j-th column are removed.

Example of the Laplace expansion according to the first row on a 3x3 Matrix.

$\mathrm{det\; A}= \begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}$

The first element is given by the factor a₁₁ and the sub-determinant consisting of the elements with green background.

$\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array} =>a_{11} \begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}$

The second element is given by the factor a₁₂ and the sub-determinant consisting of the elements with green background.

$\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array} =>a_{12} \begin{array}{cc}{a}_{21}& a_{23}\\ a_{31}& a_{33}\end{array}$

The third element is given by the factor a₁₃ and the sub-determinant consisting of the elements with green background.

$\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array} =>a_{13} \begin{array}{cc}{a}_{21}& a_{22}\\ a_{31}& a_{32}\end{array}$

With the three elements the determinant can be written as a sum of 2x2 determinants.

5x5x5 Shipping Boxes

$\mathrm{det\; A}= \begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array} =a_{11} \begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array} -a_{12} \begin{array}{cc}{a}_{21}& a_{23}\\ a_{31}& a_{33}\end{array} +a_{13} \begin{array}{cc}{a}_{21}& a_{22}\\ a_{31}& a_{32}\end{array}$

It is important to consider that the sign of the elements alternate in the following manner.

$\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}$