Shifting Times Tables

This problem encourages you to think about the characteristics of numbers and the times tables that they are part of. Noticing these patterns is not only fun, but will also help you with calculations in other work that you do.

Many great solutions were submitted, with clear explanations, so thank you very much to everyone for this!

Let us first consider levels one and two. In these levels, the numbers displayed are always the first five numbers from a shifted times table. A student from Mearns Castle High School explained:

In Levels $1$ and $2$, if the first number is x and the second is $x+d$, then the d times table has been altered.

Using this notation, we can also find an expression for the third number: $x+2d$. In fact, we can write a formula for any number. For example, to find the $n^{th}$ number, we can write a general formual: $x+(n-1)d$. Note that in levels one and two, the times table number is the same as "d". However, this may not be the case in levels three and four, as the numbers may not be consecutive.

The question also asks you to work out the number that the times table has been shifted by. Holly, Nick, and Lucy from Old Earth Primary School first found the times table. Then they subtracted this number (e.g. "$5$" for the five times table) from the first number of the sequence to find the shift. Sharumilan from Wilson's School, and James from Woodstock CE Primary School used a similar method.

When solving the problem, several students found it helpful to look at the properties of the numbers in the sequence. For example, they noted whether the units digits were all odd, or all even, or a mixture, and whether they were all identical or whether there were only two different digits. This can eliminate some possibilities for the times table and shift, whilst pointing you in the right direction to find the answer. Other characteristics students examined included the type of difference between the numbers in the sequence (prime or composite).

Emma from Tadcaster Grammar School answered the questions posed regarding the Level 1 and 2 problems:


1. WHAT CAN YOU SAY IF THE NUMBERS ARE ALL ODD?
The times table is always even, and it has been shifted by an odd number.

2. WHAT ABOUT IF THEY ARE ALL EVEN?
The times table is always even, and it has been shifted by an even number.

3. OR A MIXTURE OF ODD AND EVEN?
The times table is always odd; the shift could be odd or even.

4. WHAT CAN YOU SAY IF THE UNITS DIGITS ARE ALL IDENTICAL?
The times table has a zero as its unit digit i.e. it is a multiple of ten.

5. WHAT IF THERE ARE ONLY TWO DIFFERENT UNITS DIGITS?
The times table has a five as its unit digit.

6. WHAT CAN YOU SAY IF THE DIFFERENCE BETWEEN TWO NUMBERS IS PRIME?
The times table is prime.

7. WHAT CAN YOU SAY IF THE DIFFERENCE BETWEEN TWO NUMBERS IS COMPOSITE (NOT PRIME)?
The times table is composite.

Many others submitted correct answers to these questions. These include: Sharumilan from Wilson's School, Harry from Maidstone Grammar School, Stephen and Alexander from Tudhoe Grange, Andrew from Charters Secondary School, and Jack from Sir Harry Smith Community College. Several mentioned the importance of considering factors and multiples, in order to come to the correct conclusions about the times tables. Emma mentioned this in her solution (see above).

Harry from Maidstone Grammar School explained an approach that would work at all levels:

Can you explain how you worked out the table and shift each time, and why your method will always work?

1. Put the numbers given in ascending order.
2. Find the difference between each number and the number next to it.
3. Find the highest common factor of all the differences (this is the table)
4. Divide one of the numbers by the highest common factor and find the remainder (this is the shift).

My method will always work because the differences between each number must be a multiple of the table, or the table itself, as otherwise you would not be able to increase the original number to the next.

e.g. 43 --- 55 --- 61 The difference between 43 and 55 is 12, the difference between 55 and 61 is 6, therefore the highest common factor is 6. 43 divided by 6 = 7 remainder 1 so the shift is 1!

This is a lovely explanation, and a very good systematic way to approach the problem. Many students, like Harry, mentioned that the first step is to order the numbers. This is a useful start to very many problems, not just in mathematics, but in other subjects too. It is helpful to begin by logically organising the information that you are given so that patterns can be seen more clearly and the information can be manipulated more easily.

For levels 3 and 4, the numbers given can be any five numbers; they are not necessarily in order. This means that the difference between two of the numbers may not be the actual number of the times table; there may be other numbers in between these, which have not been provided. This is why we need to find the highest common factor of all of the differences. Some students just used the smallest difference between the numbers in the sequence and concluded that this is the times table number. However, this difference may in fact be a multiple of the "true" difference.

Here is a solution that illustrates these points nicely. It was submitted by Luke, from Sawston Village College who participated in a Maths Masterclass:

For Levels three and four, where the numbers are in a random order, it is advisable to look at all of the differences, as sometimes they are different.

For example: $239, 459, 519, 579, 619$.

In this, the smallest difference is $40$, but there are also differences of $60$ and $220$.
Because you must enter the largest times table that works for all of the numbers, you cannot enter $40$ (the smallest difference), as that doesn't work for all the differences.
$20$ goes into all of these, so it is the largest times table that works for all of the differences.

However, if one of the differences you find is prime, there is no point in going through the process of looking at all of the differences, as it isn't divisible by anything apart from $1$ or itself, and therefore will not share a factor with any other differences you find. The shift is not altered by different differences.

Thank you also to the following students, who submitted great explanations for working out the times table and the shift: the two "Toms" from Bassingbourn Village College, Stephen and Alexander from Tudhoe Grange School, and Holly, Nick and Lucy from Old Earth Primary School.

If you enjoyed this problem and would like to follow it up, have a look at the article on Modular Arithmetic.