Differs
Choose any 4 whole numbers and take the difference between
consecutive numbers, ending with the difference between the first
and the last numbers. What happens when you repeat this process
over and over again?
Age
11 to 14
Challenge level
Problem
Choose any 4 whole numbers, for example (100, 2, 37, 59). Now take the difference between consecutive numbers, always subtracting the smaller from the larger, and ending with the difference between the first and the last numbers. What happens when you repeat this process over and over again? For example, continue the sequence of 4-tuples:
(100, 2, 37, 59), (98, 35, 22, 41), (63, 13, 19, 57), (50, 6, 38, 6) , ...
Investigate starting with different sets of 4 numbers.
What happens when you use 3 numbers, or 5, or 6, or 7 or 8 numbers?
Student Solutions
Here are some examples from Robert Dunne and Tom Antnistle of St James Middle School, Bury St Edmunds.
(50, 6, 38, 6), (44, 32, 32, 44), (12, 0, 12, 0), (12, 12, 12, 12), (0, 0, 0, 0)
(86, 74, 34, 28), (12, 40, 6, 58), (28, 34, 52, 46), (6, 18, 6, 18), (12, 12, 12, 12), (0, 0, 0, 0)
With another lovely solution from Suzanne andNisha, (Y10) The Mount School, York.
We choose any 4 whole numbers, for example (100, 2, 37, 59) and take the difference between consecutive numbers, always subtracting the smaller from the larger, and ending with the difference between the first and the last numbers. When you repeat this process over and over again:
Eg: (100,2,37,59), (98,35,22,41), (63,13,19,57), (50,6,38,6), (44,32,32,44), (12,0,12,0), (12, 12, 12, 12), (0,0,0,0)
all sets of four numbers end up as four zeros. Sometimes it takes longer than others.
Because the same term of the sequence is repeated over and over we say that the system ends in a fixed point (0,0,0,0).
Now let's try three numbers
(50,46,55), (4,9,5), (5,4,1), (1,3,4), (2,1,3), (1,2,1), (1,1,0), (0,1,1), (1.0.1), (1,1,0), (0,1,1), (1,0,1), ......
This pattern of two 1's and one zero continues for ever. The last 3 terms are repeated over and over again. They are called a 3-cycle.
(100,2,37), (98,35,63), (63,28,35), (35,7,28), (28,21,7), (7,7,21), (0,14,14), (14,0,14), (14,14,0), (0,14,14), ........
This time it's two 14?s and one zero and the pattern is slightly longer, but still goes on for ever ending in a repeating 3-cycle.
With five numbers:
(100,2,37,59,4), (98,35,22,55,96), (63,13,3,41,2), (50,20,8,39,61), (30,12,31,22,11), (18,19,9,11,19), (1,10,2,8,1), (9,8,6,7,0), (1,2,1,7,9), (1,1,6,2,8), (0,5,4,6,7), (5,1,2,1,7), (4,11,6,2), (3,0,5,4,2), (3,5,1,2,1), (2,4,1,1,2), (2,3,0,1,0), (1,3,1,1,2), (2,2,0,1,1), (0,2,1,0,1), (2,1,1,1,1), (1,0,0,0,1), (1,0,0,1,0), (1,0,1,1,1), (1,1,0,0,0), (0,1,0,0,1), (1,1, 0,1,1), (0,1,1,0,0), (1,0,1,0,0), (1,1,1,0,1), (0,0,1,1,0), (0,1,0,1,0), (1,1,1,1,0), (0,0,0,1,1), (0,0,1,0,1), (0,1,1,1,1), (1,0,0,0,1), (1,0,0,1,0), ......
Again this ends in a repeating cycle of period 15. Note that (1,0,0,0,1) is repeated so the cycle of 15 terms starting with (1,0,0,0,1) is repeated over and over again.
Some are quick: (5,12,26,7,1), (7,14,19,6,4), (7,5,5,5,3), (2,0,0,2,2), (2,0,2,0,0), (2,2,2,0,2), (0,0,2,2,0), (0,2,0,2,0), .... repeating 2's and zeros.
Nisha and Suzanne say "The only ones we could find with all zeros was with four numbers. There are all sorts of patterns in these others, but it's not easy to generalise them.''
Try 8 numbers and see what happens. Is it true that with 3 numbers you always end with a 3-cycle, with 5 numbers you always end with a 15-cycle and with 6 numbers you always end with a 6-cycle?