Completing and checking a Magic Square
This guide has two main purposes:
-
to introduce magic squares and encourage you to investigate them;
-
-
to introduce some important techniques for constructing and
entering spreadsheet formulae.
Index
Double-click an entry to go directly to that step.
Begin with step 1!
1) Setting-up
2) Magic Squares
3) A diagonal total
4) A column total
5) Auto totalling
6) Auto solo
7) A row total
8) Copying cells
9) The other diagonal
10) Copying cells again
11) New squares for old?
12) Solve these!
1. Magic Squares
A square of numbers in which the columns, rows and
diagonals have the same total is called a magic
square. The shared total is called the square's magic
number.
Copy the magic square begun here into the block of
bordered cells, and then complete the square. Begin
by finding the square’s magic number from the
completed diagonal which runs up from cell D5 to F3.
4
7
6
3
A diagonal total
You will have found that placing numbers in a magic
square involves lots of mental arithmetic - easy, but
far too much of it for comfort! Why not make the
spreadsheet do the work? Here's how ...
Cells display numbers or words. However, they can
contain numbers, words, or formulae.
A formula is simply a rule for calculating what
number (or words) should be displayed.
Enter a formula to total the numbers in the diagonal
from D3 to F5 as follows:
-
click into cell G6
-
beginning with the equals sign, enter the
formula:
= D3 + E4 + F5
-
complete the entry by tapping <Enter> ...
the formula is at once evaluated (worked out)
and the result displayed
4
A column total
Now click into cell D6 and enter a formula to total
the numbers in the column from D3 to D5. You will
need to use what you learned in step 3 - remember
to begin with an equals sign.
5 Auto totalling
5
Provided a block of cells is already filled with
numbers, you can ask Excel to write a totalling
formula for you! Try this on the column from E3 to
E5, as follows:
-
click into cell E6
-
click the AutoSum button
<Enter>
-
look at the formula line (just above the
letters at the top of each column) and you will
see the formula:
-
and then tap
= SUM(E3:E5)
... this simply means:
Find the SUM of
the block of cells which begins at E3 and ends
at E5
6
Auto solo
Now get Excel to enter in cell F6 a formula to total
the last column. (Refer to the previous step if you
need a reminder on the method!)
7
A row total
In step 5 you learned that Excel can be made to
write a SUM formula for any block of cells
containing numbers - the AutoSum button works for
rows exactly as it does for columns.
Get Excel to enter in cell G3 a formula to total the
first row of the magic square
8
Copying cells
You could use the method of step 7 to enter
formulae for the two remaining row totals in cells G4
and G5. However, Excel can be made to use the one
formula you have entered as a pattern for creating
the rest automatically. Here's how:
-
click into cell G3
-
move the pointer to the bottom-right corner
of cell G3, so that the pointer looks like
cross-hairs ...
... and then drag the corner of the cell down
the column to cell G5
-
have a look at the formulae written by Excel
by clicking into cell G5 …
…on the formula line you will see the formula:
= SUM(D5:F5)
... Excel has adjusted the copied formula so
that it does the same job relative to its
surroundings (that is, so that it adds up the
three cells to its left)
-
9
if you click into cell G4 you will see that the
formula there has been adjusted in the same
way
The other diagonal
It remains to enter in cell G2 a formula to total the
other diagonal (cells D5 to F3). The AutoSum button
works only for a block of cells in one row or one
column: for this job you will need to use the method
of step 3 - get there via the Index if you would like
a reminder.
10 Copying cells again
There are other arrangements of the numbers you
have placed in the magic square which also provide a
solution. And there are other sets of numbers which
can provide solutions. To explore these it will be
useful to have a second square, complete with all the
totalling formulae. You could start from scratch but copying can save you the trouble! Here's how:
-
select your magic square "kit" by dragging
from cell D2 to cell G6
-
click the Copy button
-
click into cell D10 and click the Paste button
-
try changing any number in the second square
... the affected totals change correctly
-
see how Excel copied the formulae by clicking
into cell G11: the formula bar displays:
= SUM(D11:F11)
... the formula copied from cell G3 has been
adjusted so that it does the same job relative
to its surroundings
-
look at the formulae in F14 and G14 to check
that the same thing has happened to
formulae for column and diagonal totals
11 New squares for old?
Using your second magic square kit, see what other
successful arrangements you can find of the
numbers placed in the first square.
Are some of them really different, or are they just
different ways of looking at your first square? (One
other way of looking at your first square would be to
stand on your head - or to turn the "paper" upside
down).
12 Solve these!
Using your second magic square kit (or further
copies of the kit created by copying and pasting as
before), see if you can complete these magic
squares:
(a)
(b)
8
3
11
6
12
4
15
(c)
13
(d)
10
6
5
13
6
7
13
8
14
9
15
That’s it!
Congratulations! You have completed this guide.