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14 Marts 2007 @ 18:17
Railway sequences  

Not long ago, I had occasion to travel to Ottawa and back on VIΛ (which is the only Crown corporation I know of that's named after a preposition, and which spells its name with a lambda instead of an A presumably for the sake of rotational symmetry), so I had a look at the puzzles near the back of the February/March issue of Destinations, their earthbound equivalent of an in-flight magazine. Unlike some, I'm no puzzle expert,1 but I do have a couple of critical thoughts to share from my decidedly amateurish perspective.

Spoiler alert: If you are planning to ride a train in Canada before the end of this month (or whenever VIΛ gets around to distributing the next issue of Destinations), then reading the rest of this post may deprive you of a few brief moments of mildly pleasant diversion on your journey. You may find yourself compelled to spend those minutes looking at the passing scenery instead.2 You have been warned!

The puzzles I was interested in all involved finding the next item in a given sequence. Three of them were sequences of integers:

  • 6-8-12-20- ?
  • 2-6-14-30- ?
  • 4-5-7-11-19- ?

Any one of these I would have found uninspired but tolerable by itself, but for them to print all three in the same issue just seemed silly. What do you think?

Another sequence involved pairs of integers and letters:

  • 3:T, 6:S, 7:S, 9:N, 13: ?

This isn't very original, but eventually I realized that it's actually at least a little bit cleverer than it looks. You can make a more interesting puzzle out of it by asking for the next number–letter pair, rather than giving the number and asking which letter goes with it.

Here are VIΛ's answers and explanations for the first set:

  • 36 (–2 ×2)
  • 62 (+1 ×2)
  • 35 (×2 –3)

If you look at it like that, then each of them has a different method for obtaining the next number in the sequence, although each one involves multiplying by two. But I was solving these by looking at the differences between adjacent numbers (which is what I usually try first in puzzles like this), and in each one, you just keep adding the next power of two. So to me, this was essentially the same puzzle three times over, rather than three different puzzles—the numbers of the overt sequences sort of fade away into the background, and the powers of two between them stand out. (I'd be interested to hear if my readers had the same reaction, or if you see these sequences differently.)

In the second one, Thirteen maps, of course, to T. The next pair would have to be 16:S. The numbers given by VIΛ look like the beginning of this sequence, but the railway's choice of numbers had a linguistic basis rather than a mathematical one. The puzzle is bilingual, so the sequence of numbers is the ones that begin with the same letter in English and French:

3679131630...
threesixsevenninethirteensixteenthirty...
troissixseptneuftreizeseizetrente...

(And they could have started off with 0:Z, but that might have been too obvious.)

This is, I think, a pretty neat exploitation of the similarity between Canada's two official languages.


1. My one tenuous basis for a claim of authority here is that I was once invited to co-edit a scholarly journal on enigmatology. I declined on the grounds that I was wholly unqualified.

2. If you encounter delays en route, you may even be compelled to watch non-passing scenery.

 
 
 
love, play & inquirytrochee on 14. Marts, 2007 22:24 (UTC)
yep, I agree that the first triple-question all looked like the same problem to me: adding subsequent powers of two. Differences are almost always the place to start in sequences, for me.

and I agree that the linguistic justification is pretty neat for the later puzzle -- and regardless of starting in French or English, 13:T.

It reminds me of the following number sequence:
3, 3, 5, 4, 3, 5, 5, 4, 3

but I probably give myself away, unless I made an error.
Q. Pheevrq_pheevr on 14. Marts, 2007 22:37 (UTC)

I guess the VIA version of that one would have to go something like this:

0:4, 3:5, 5:4, 6:3, 9:4, ...
Merlemerle_ on 15. Marts, 2007 00:52 (UTC)
I saw the sequences in exactly the same way you did: powers of two. It seemed quite obvious. It's much easier to think of solutions involving one operation, rather than two. They could at least have multiplied by three, or subtracted by three or more, to prevent them all from having the same pattern.

But I have an issue with problems like that. Honestly, the answer to all of them could be 0, 13, -496, or any other number. Given n terms I can (given time) construct an order n+1 polynomial such that the next term is whatever number I want. They need to phrase the puzzle better, to ask you to find the simplest next number that fits a pattern which requires the least complicated computation. (of course, then their answer would have been a failure, but..)

That second puzzle was indeed interesting, though! It seemed a bit trickier than the level of puzzle one normally finds in publications by transportation agencies.
Q. Pheevrq_pheevr on 15. Marts, 2007 17:56 (UTC)
But I have an issue with problems like that. Honestly, the answer to all of them could be 0, 13, -496, or any other number. Given n terms I can (given time) construct an order n+1 polynomial such that the next term is whatever number I want. They need to phrase the puzzle better, to ask you to find the simplest next number that fits a pattern which requires the least complicated computation. (of course, then their answer would have been a failure, but..)

I don't think it's possible to reason inductively without at (at least implicitly) reasoning abductively as well. So in any continue-the-sequence puzzle, the instructions pretty much have to be taken to mean something like "find the next term consistent with the simplest hypothesis about how the sequence works." In the first example, "6-8-12-20- ?," the most elegant hypothesis seems to be that the nth term in the sequence is 2n+4. VIA describes the pattern recursively, showing how one can calculate the n+1st term given the nth; you and I saw it according to a different recursive definition in which you need the nth and n+1st terms to calculate the n+2nd. But these are just different ways of describing the same sequence, which is (as far as I can tell) the simplest one starting with the four terms given.

Merlemerle_ on 15. Marts, 2007 20:30 (UTC)
I don't think it's possible to reason inductively without at (at least implicitly) reasoning abductively as well.

That's true. Or, at least, we are not really wired to separate the various sorts of reasoning, and limiting ourselves to one type would significantly decrease the utility of reason.

However, in several instances (I do a lot of puzzle magazines), I have come up with a solution different from theirs. Both were valid solutions, and in some cases one was simpler than the other, but in others they were very similar. It disturbs me when they give a "definitive" answer that is no better than another answer was.

But you are correct, I was using that frustration to create a rather extreme example.
O.K.: king of the mountaincaprinus on 15. Marts, 2007 01:17 (UTC)
Wait, that first bit, you're joking, right? Via = road/way, and the logo is just silly typography... ?
Q. Pheevr: Cello case sketchq_pheevr on 15. Marts, 2007 18:12 (UTC)

Well, I'm joking about the lambda, at least; in all contexts other than the logo, VIA spells its name with an A, so it's probably fair to assume that the Λ in the logo is really a barless allograph of A rather than a lambda.

As for the meaning of the name... well, via is a preposition in both of the languages in which VIA Rail Canada does business. That it was a noun in Latin probably figured into the calculations of the people who picked it as a name, but who knows?

Henrytahnan on 15. Marts, 2007 01:46 (UTC)
Wow. Yes, the "differences of powers of two" kind of leapt out at me too. On the other hand, I'm just not sufficiently bilingual to have seen the pattern in the other puzzle, which is indeed kind of neat.
Eryneryn_ on 22. Marts, 2007 17:47 (UTC)
I had the same reaction to the sequences of integers puzzles. "Wait, we just had this answer." Normally I am not adept at those kinds of puzzles as they are not interesting at the lower levels and at higher levels tend to involve math I cannot do in my head.

I didn't get the letter puzzle at all. Whenever there are those puzzles, I always feel stupid afterward.

How was the trip otherwise?