## Wednesday, September 07, 2011

### Next in the sequence

When I posted on sequence problems here some people said that they did not have unique answers. Many problems FORMALLY do not have a unique answer, but common sense and simplicity lead to a unique answer. This recently happened on Jeopardy. The topic was Next In The Series. I'll give the questions here and pointers to the answers (actually the answers here and pointers to the questions since Jeopardy is an answer-and-question game.) Do you think some of these have non-unique solutions? If so, what are the other solutions? Are those problems unfair? As always I ask nonrhetorically. (my spell checker wanted me to put a space between non and rhetorically. I disagree.)
1. FA, SOL, ... NEXT
2. In the US Army: First Lt, Captain, ... NEXT
3. XXXVIII, XXXIX,... NEXT
4. FEDERAL HOLIDAYS: Memorial Day, Independence Day, ... NEXT
5. Orange, Yellow, (Wavelength around 510 nanometers)... NEXT

1. 4 could possibly be followed by either Nakba Day or Patriot Day (depending on how you might want to interpret FEDERAL HOLIDAY)

2. If the numeric sequence were 1,2,3,..., I doubt there would me much debate over the next number.

I think there is no doubt that n++ or "Item.Successor()" is that simplest of all possible algorithms that could define a sequence.

3. I think they're all reasonable, but I also think your sequence puzzles are reasonable. (I don't subscribe to the usual pedantry.)

4. Bill, these are not mathematical questions and that is the point! Yes, if you ask what is the word that most often comes next after A, B, C, D then the answer is E, but it is NOT a mathematical question. If you give me a file full of finite sequences of numbers and then give me a partial sequence that appears in one or more of them and ask me what is the next number I can compute the most likely next element.

For comparison, what is the next sequence of characters in the following sequence:

SHRDX, GTDFDS, ODLQDG, MVNDYE, QPXLGR, ?

5. In the world of color printing, any sequence that begins "orange, yellow" is a logical train-wreck right from the start.

This is because on a four-color press (that is, a press with the four ink colors cyan, magenta, yellow, black) the color "yellow" is easy to render, whereas a saturated "orange" is out-of-gamut, that is to say, impossible-to-render.

High-quality printing therefore requires special presses with larger numbers of inks; as I recall the books of Edward Tufte (author of Visual Display of Quantitative Information) are printed on six-color or possibly even eight-color presses.

The bottom line is that for artists and craftspersons in general, and printers in particular, no color sequence is logical, until the details of the color reproduction method have been clearly stated.

6. Nobody in their right mind would answer Nakba Day in this context, and while it is not a holiday in any sense and is officially listed (as many things are) as a Federal day of observance, Patriot Day is after Labor Day.

7. 1,2,3 could easily be followed by 5

8. Math grad student formerly employed by a printer1:44 PM, September 08, 2011

@John Sidles

What you say is true for printing (or subtractive color), but the mention of wavelength in a Jeopardy! clue immediately places it in the context of light (or additive color), specifically of increasing wavelength.

Even so, when does yellow end and some other color begin? What naming convention are using? Of course, this is a part of Jeopardy! and I doubt Ken Jennings, Brad Rutter or Watson would complain about these minutia.

9. Oh I entirely agree with you, MathGradStudent. Arguably, a Watson that could compose good Jeopardy questions (which often are witty or punning) would be far more human-like than a Watson capable merely of answering said questions.

Similarly, an AI that could compose mathematical definitions that led naturally to good theorems, would be exhibit a higher order of mathematical intelligence than one that proved those theorems.

As Michael Spivak has written:

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There are good reasons why theorems should all be easy and the definitions hard … Definitions serve a twofold purpose: they are rigorous replacements for vague notions, and machinery for elegant proofs.

Stokes' Theorem shares three important attributes with many fully evolved major theorems: (1) It is trivial. (2) It is trivial because the terms appearing in it have been properly defined. (3) It has significant consequences.
---------------------

Mathematical luminaries who have written upon the key role of good definitions in proving good theorems include Grothendieck, Thurston, and Atiyah … I will leave it to other folks to supply the quotations! :)

10. http://spikedmath.com/062.html