Shifting Times Tables
1. How to find out the times table (T) and shift (S)
Levels 1 + 2
T = difference between any two (of the five) consecutive numbers
S = first number - difference between any two consecutive numbers
Levels 3 + 4
T = difference between two closest numbers
For example, if my numbers were 59, 72, 85, 98, 137 (in order), the times table would be 13 because the two closest numbers of these 5 are 72 and 85, and the difference between these is 13.
To find S, take the time table root repeatedly from the lowest number until you get the lowest number possible that is still higher than the time table root. The difference between this number and the root is S. For example, in the example above, I would take 13 repeatedly from 52 until I could no longer take 13 away without the number becoming smaller than 13 itself (this number is 20). The difference then between 20 and 13 is 7, therefore S = 7.
If the lowest number is below the times table root, then the shift is negative. If your numbers were -12, 16, 30, 37, 58, then you would take the difference between the two closest numbers (30 and 37, which in this case is 7, and add it to the lowest number until you got the highest possible number without going higher than the times table root (7). The amount you have to add to the root to get this number (2) is S (note that this will always be a negative number - in this case it is -5)
2. What if the numbers are....
All odd? - This means that one of the values (t or S) is even and the other is odd
All even? - This means that both T and S are odd/even
Odd and even? - This means that both T and S are odd
3. What if the units' digits are all identical?
If they are all identical, eg. 12, 22, 32, 42, 52, then T must be 10.
What if there are only 2 different units digits?
The only possibility for this to happen would be if the two units digits were 5 and 0, in which case T would have to be 5 or a number ending in 5 (eg 15, 25, etc).
4. What if the difference between two numbers is prime?
If they are two consecutive numbers in the altered times table, then T is that number and therefore prime.
What if the difference between two numbers is composite?
Then T is composite.