To start, let's establish some rules about times tables:
If the times table is even (say the two times table) all numbers in the times table will one of even or odd (they will not rotate between the two) (1)
If the times table is odd, the numbers could end in either an odd or even number (2)
If the times table is a multiple of 5 then it will always end digits that are one of two numbers (3)
If the times table is a multiple of 10 then it will always end in the same digit (4)
Let's look at an example:
348, 92, 252, 284, 124
From (1) and (2) we can see that the times table is even. We can also see that this sequence breaks (3) and (4) so therefore cannot be a multiple of either 5 or 10.
To work out exactly what the times table is we need to subtract the lowest number by the second lowest number. In this case it is 124-92 which equals 32. From this we can tell that the times table must be a factor of 32.
We then need to subtract the third lowest number by the second lowest number (252-124=128), the second highest number by third lowest number (284-252=32) and the highest number by the second highest number by the highest number (348-284=64)
This gives us four numbers: 32,128,32 and 64. We need to work out the highest common factor for these numbers - in this case it's 32. From this we can see that the times table is 32 as the numbers are a multiple of 32 apart.
To find the shift up or down we need to divide one of the numbers in our times table and look at the remainder. In this case it is 92/32=2 r28. From this, we can see that if we were to move our number down 28 it would be a multiple of 32. This gives us our solution - the times table is 32 and you need to move it down 28.
In algebraic form we could write the way to find the times table like this:
If a = the highest number, b = second highest number (and so on)
T (Times table) = HCF(Highest common factor) a-b, b-c, c-d, d-e
To find the shift we could do this:
If t = times table and a = highest number
S(Shift down) = a - remainder a/t
Thank you for reading my solution