Tables Without Tens
Katie from Sir Jonathan North School made the following
remark:
"In the rows of odd numbers, except $5$, you have every number from
$1$-$9$, and in the rows of even numbers you have the multiples of
$2$. This occurs again in the columns. For example, (ignore the
presence of the columns of zero's for a second) in the first
column, third column, seventh colum and ninth column, every number
from $1$ to $9$ is present."
Why do you think the rows and columns are the same?
A number of you remarked that the table is symmetrical across the
diagonals. Can you explain why this is?
Katie's first remark is the most interesting - we get the numbers
$1$ - $9$ in any row that is not a multiple of $2$ or $5$. Does it
surprise you that these are the factors of $10$?
Extension
Matthew from QEB correctly observed that behind this problem is an
important kind of arithmetic called 'modular' or 'clock'
arithmetic. A search for either of these terms on the NRICH website
should yield some interesting problems and articles on this topic.
For an article aimed at secondary school pupils click here
.