# I need a new hobby, what can I do that's productive and fun?

I have a lot of free time, any suggestions?

0|0
410

• You are a Mathlete.

Try to solve this problem:

Let X (k) be a sequence of positive integers defined here:
X (k+1) = 3 * X (k) + 1 if X (k) is odd.
X (k+1) = X (k) / 2 if X (k) is even.

Show that, for any starting value of X (0) that is a positive integer, the sequence will eventually become {...,4,2,1,4,2,1,4,2,1,...}

For example:
X (0) = 7
7 is odd, so
X (1) = 3 * X (0) + 1 = 3 * 7 + 1 = 21 + 1 = 22
22 is even, so
X (2) = X (1) / 2 = 22 / 2 = 11
11 is odd, so
X (3) = 3 * X (2) + 1 = 3 * 11 + 1 = 33 + 1 = 34
34 is even, so
X (4) = X (3) / 2 = 34 / 2 = 17
17 is odd, so
X (5) = 3 * X (4) + 1 = 3 * 17 + 1 = 51 + 1 = 52
52 is even, so
X (6) = X (5) / 2 = 52 / 2 = 26
26 is even, so
X (7) = X (6) / 2 = 26 / 2 = 13
13 is odd, so
X (8) = 3 * X (7) + 1 = 3 * 13 + 1 = 39 + 1 = 40
40 is even, so
X (9) = X (8) / 2 = 40 / 2 = 20
20 is even, so
X (10) = X (9) / 2 = 20 / 2 = 10
10 is even, so
X (11) = X (10) / 2 = 10 / 2 = 5
5 is odd, so
X (12) = 3 * X (11) + 1 = 3 * 5 + 1 = 15 + 1 = 16
16 is even, so...
(In fact, it's a power of 2, so...)
X (13) = X (12) / 2 = 16 /2 = 8
X (14) = 4
X (15) = 2
X (16) = 1
1 is odd, so
X (17) = 3 * X (16) + 1 = 4

So, just to summarize the sequence:

7 -> 22 -> 11 -> 34 -> 17 -> 52 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 -> 4 -> 2 -> 1 -> 4 -> 2 -> 1 -> ...

It so happens that, no matter what positive integer you start with, you end up with...
... -> 4 -> 2 -> 1 -> 4 -> 2 -> 1 -> 4 -> 2 -> 1 -> ...

Why is that?

Prove it.

0|0
0|0
• I appreciate the time you took to find this or write it, but this is not at all what I had in mind

• The 'fun' part depends on what sort of person you are. Running, wall climbing, dancing, gardening or cooking/baking may be fun. You could also pick up something creative like painting/drawing.

0|0
0|0