# Slot Machine Challenge

Karen owns a restaurant/bar. She wants to attract more customers, and is building her own custom slot machine game, which she can use to give away free drinks or prizes. The more a customer spends, the more spins they get on the slot machine.

She knows that she wants these specs:

1. 4 reels on the machine
2. All reels must have a different number of symbols, between 2 and 9 inclusively (what the symbols are is irrelevant for this challenge)
3. All reels must move (change to another symbol) during a spin.
4. All of the symbols on each reel must be unique on that reel.

To avoid using a random number generator, she decides to set each reel to cycle uniformly through each of its possible positions. For example, if a reel has 4 symbols, say, 1,2,3,4, then it will cycle through them in the same pattern, say, 4,2,1,3,4,2,1,3,....

She knows that this guarantees that the machine will cycle through all of its possible states in the exact same order each cycle, but decides that that is OK. She desires to make this cycle as long as possible, to make the machine feel more unpredictable, but she has no idea how to find the correct combination of reel sizes. Although, she does understand that the order of the reels does not matter.

Below is a generic picture of slot machine reels that I found on Google images. It is added only so you get a general idea of slot reels, and is not directly related to this puzzle.

Her friends all have an opinion, and give the following advice about the number of symbols for each reel:

Anna: 6,7,8,9
Brett: 4,5,7,9
Candice: 5,6,8,9
Daniel: 4,5,6,7
Eileen: 2,7,8,9
Fred: 3,5,7,9

Questions:

1. Which friend gave the best advice?

2. Which friend gave the worst advice?

3. Can you find a more optimal combination than the best advice?

The total number of states is the LCM of the numbers of symbols on each reel. So following the four friends' advice, she would have the following total number of states:

Anna: LCM(6,7,8,9) = 504
Brett: LCM(4,5,7,9) = 1260
Candice (weird name btw): LCM(5,6,8,9) = 360
Daniel: LCM(4,5,6,7) = 420
Eileen: LCM(2,7,8,9) = 504
Fred: LCM(3,5,7,9) = 315

So Brett gave the best advice while Fred gave the worst. The answer to the third question is yes, by the following argument.

The LCM of four numbers between 2 and 9 inclusive is going to have all its prime factors in {2,3,5,7}. The maximum power of 2 we can have is $2^3=8$; the maximum power of 3 is $3^2=9$; of 5 and 7, just $5^1=5$ and $7^1=7$. So the optimal combination is 5,7,8,9, giving total number of states LCM(5,7,8,9) = 5*7*8*9 = 2520.

• rand where are you from? you've never heard candice before? – JLee May 10 '15 at 22:40
• also, do you think this puzzle is too simple / uninteresting? in other words, do you think there would be an audience that would enjoy it, or is it too basic? if i was to delete it, would you lose your points for it? – JLee May 10 '15 at 22:50
• @JLee I'm from Britain, and have heard Candy but not Candice... – Rand al'Thor May 10 '15 at 22:52
• @JLee I found it very simple (the only challenge was to extract the essential maths of it from the context of the puzzle), but non-mathematicians might find it more interesting/challenging. And yes, if you deleted it I'd lose 25 points for this answer! – Rand al'Thor May 10 '15 at 22:54