The Riddler - Series on FiveThirtyEight

[TheRealJoeFish]

The Riddler PUZZLE THREE - How Long Will Your Smartphone Distract You From Family Dinner?

This is the 3rd puzzle in FiveThirtyEight's The Riddler series, in which Oliver Roeder posts an interesting math/creative thinking-ish brain-teaser every Friday.  I'll post either my solution (if I can come up with one) or my initial thoughts (if I don't see how to solve it) in hide tags, and everyone is welcome to give it a go! I have promised a rep to people who solve a problem that I got wrong

I'm one for two; as noted above, I crashed and burned on the first one but was able to brute force the second.  I haven't looked at this one yet, so, let's see if this will join the "stumps Joe" category:

Now, here’s this week’s Riddler, which comes to us from Olivia Walch, a mathematics Ph.D. student and cartoonist:

You’ve just finished unwrapping your holiday presents. You and your sister got brand-new smartphones, opening them at the same moment. You immediately both start doing important tasks on the Internet, and each task you do takes one to five minutes. (All tasks take exactly one, two, three, four or five minutes, with an equal probability of each). After each task, you have a brief moment of clarity. During these, you remember that you and your sister are supposed to join the rest of the family for dinner and that you promised each other you’d arrive together. You ask if your sister is ready to eat, but if she is still in the middle of a task, she asks for time to finish it. In that case, you now have time to kill, so you start a new task (again, it will take one, two, three, four or five minutes, exactly, with an equal probability of each). If she asks you if it’s time for dinner while you’re still busy, you ask for time to finish up and she starts a new task and so on. From the moment you first open your gifts, how long on average does it take for both of you to be between tasks at the same time so you can finally eat? (You can assume the “moments of clarity” are so brief as to take no measurable time at all.)

Hmmm..............................  Initial thoughts below in hide tags, then I'll open the floor to discussion before I post an attempted solution or an admission of defeat!

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Interesting.  At first glance, this looks similar to Problem 2 in that it appears to be solvable by multiplying probabilities, to get P(1 Minute) + 2*P(2 Minutes) + 3*P(3 Minutes) + ... and so on.  Of course, we may need to use some sort of trick to calculate that infinite sum.  Hopefully, it will simplify to a very straightforward geometric sequence, so we can just add it up using (1 + n + n^2 + n^3 + ...) = 1/(1-n).

Another way of looking at it, which might be easier if the above math doesn't come easily or obviously, would be to look at the probability of changing between states.  There are essentially nine different states:

1) You're clear but your sister has 1 minute left;
2) You're clear but your sister has 2 minutes left;
3) You're clear but your sister has 3 minutes left;
4) You're clear but your sister has 4 minutes left;
5) You have 1 minute left but your sister is clear;
6) You have 2 minutes left but your sister is clear;
7) You have 3 minutes left but your sister is clear;
8) You have 4 minutes left but your sister is clear;
9) You're both clear.

If we view it this way, we'll be trying to figure out the probability of a given state 1-8 transitioning into state 9, and then adding up probabilities of that happening.

The ways of thinking about it above are, of course, similar, though one sort of moves minute-by-minute and the other moves task-by-task.  The minute-by-minute approach seems more natural, and is what I'll try first, but I'm not sure which will eventually yield the answer.  If we get an answer with one, it would behoove us to see if that makes sense when viewed through the lens of the other.

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So, that's my thoughts on that! Time to put pen to paper.  THIS GOES FOR YOU ALL TOO.  REMEMBER, there is a SHINY NEW REP for someone who solves a problem I get wrong!!!

Good huntin'!