An Example Start with 3, say this gives 3 as the digit, cubing 3 gives 27, The new number is 27 this gives 2 and 7 as the digits 8 and 343 as the cubes, 8 + 343 = 351. The new number is 351 this gives 3 and 5 and 1 as the digits, 27, 125 and 1 as the cubes, 27 + 125 + 1 = 153 again giving 1 and 5 and 3 as the digits, so 153 is a fixed point.

Play around with some examples by hand and then make a guess - an educated, thoughtful guess.

Does the algorithm hit a cycle or a fixed point? Always?

And how many different cycles and fixed points do you predict there might be?

By Computer

Time to get out the computer - and check it out further.

Again, if using a spreadsheet or programming language, the MOD function, as explained on Separate and Square, may be very useful.

Once you have things working and have explored the results of applying this algorithm for a suitable range of starting numbers, answer the following questions:

How many different cycles and fixed points are there?
What are they and when do they occur?

Don’t forget to attach any notes, printouts or program listings to your response. They will be of great help in identifying what you have done.