Question

Alan and Bob are playing with numbers. Starting with the number n=1 they take turns performing arithmetic manipulations with n. Alan goes first and always multiplies n by 3. Bob adds 1 or 2 to the number n, depending on a coin toss. A number is called reachable, if it can possibly appear in this game. How many reachable numbers are there between 1 and 1000 (inclusive)?

Answer 1

3
4,5
12,15
13,14,16,17
39,42,48,51
40,41,43,44,49,50,52,53
120,123,129,132,147,150,156,159
121,122,124,125,130,131,133,134,148,149,151,152,157,158,160,161

363,366,372,375,390,393,399,402,444,447,453,456,471,474,480,483

32 numbers, all above 363, so the next step will exceed 1000

Looks like 1+2+2+4+4+8+8+16+16 = 61 numbers

I guess you could show that there are never any duplicates in the list (since adding 1 or 2 to a multiple of 3 spreads out the results enough)

Then show the number of reachable numbers at each step,
then show when the minimum number at a step will exceed 1000.

Not quite sure what principle was to be studied here.

Answer 2

oops. forgot to add in the 32 for the last line