Reasoning about the number of matches needed to build squares that
share their sides.
We can arrange dots in a similar way to the 5 on a dice and they
usually sit quite well into a rectangular shape. How many
altogether in this 3 by 5? What happens for other sizes?
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
Those who submitted solutions found that there
are many different and interesting patterns to be found in this
problem. In addition, some tried to find explanations for these
patterns, which was great. As well as being fun, this problem
should also help with learning the times tables, as a student from
Devagiri CMI Public School, Calicut in Kerala, India found out.
Firstly, let's look at the symmetrical
patterns that a few of you found in the tables.
The numbers along the diagonals of the full,
completed square show a pattern, as these people discovered: Katie
from Sir Jonathan North CC, Isabel and Eleanor from The Manor Prep,
Veronika, Miriam, Albin, Carolien and Erin from the Independent
Bonn International School, Nitya Andrew and Sydney from Ysgol Pen Y
Bryn School in Wales, and a student from Devagiri CMI Public
Katie found that the diagonal lines follow the
rule of symmetry. Isabel and Eleanor described this:
Nitya, Andrew and Sydney mentioned the pattern
using numbers to illustrate:
Veronika, Miriam, Albin, Carolien and
Erin from the Independent Bonn International School noticed
"symmetry" with the rows for three and seven. They pointed
The student from Devagiri CMI Public
School also found this.
A few people also found a symmetrical
pattern when loooking at the five times tables. Here, the only
digits in the table are $5$ and $0$. These numbers form a
cross-like pattern, running vertically and horizontally through the
middle of the table. The repetition of "$5$" and "$0$" should help
with remembering the five times tables: the "member of the table"
always ends with either a $5$ or $0$ and these alternate. Also, it
can show you quickly if a number can be divided by five with no
remainder: if this is true, the number must end in a "$5$" or
Sam from the Orchard Community Primary School
looked at patterns along the columns and rows:
A student from North Walsham Junior
School found this too, as did Nitya, Andrew and Sydney from Ysgol
Pen Y Bryn.
Luke from Tudhoe Grange found another pattern,
but this time it is along the edges of the square. He discovered
that in the first column and the last column (ignoring the columns
of zeros), there are all of the numbers from $0$ to $9$.
Furthermore, the numbers are in order: the sequence increases by
one unit going down on the left hand side, and decreases by one
unit when going down the right hand side. The same is seen for the
first and last rows (ignoring the rows of zeros).
James from Thornton Dales and James from
Tudhoe Grange found a similar sort of pattern. James from Tudhoe
Grange describes this:
Francesca and Katie went on to
Some people found patterns when looking
at odd and even numbers in the table. For example, George from
Summerswood Primary found that the first line (for the one times
table) has alternating odd and even numbers, whilst the second line
has all even numbers. For the three times table, the numbers are
alternating odd and even and for the four times table, all numbers
are even. This pattern repeats itself ...
A student from Devagiri CMI Public School also
noticed this. He explains:
He noticed this general rule: when
multiplying two odd numbers, the units are always odd. You only get
an even number if you multiply by an even number. So, to get an
even number for an answer, an even number must be involved in the
Using this rule he reasoned:
As we have seen, we can find patterns
just by looking at the numbers in the tables. Also, we can try to
find further patterns by adding, subtracting, multiplying and
dividing various sets of numbers.
Luke from Tudhow Grange and a student from
Devagiri CMI Public School looked at patterns found when adding
numbers. The student from India found complements of ten in the
table (complements are numbers that go together in some way). He
This pattern is seen for all of the
times tables. Try this for yourself, by looking along the rows of
the different times tables. Now look at the columns. Do you see the
same complements of tens?