Week 176: Computation, Space-Time, and Skynyrd
A bottle episode, not an episode where we throw bottles.
My friend Joe Wright once posed that if you want to explain the concept of space-time to someone with no interest in physics or mathematics, the concept of "four-down territory" in football, (when you need all four downs to make a score, and don’t use the fourth down to punt,) was an excellent example. See it's not entirely space (as it depends on the amount of time left in the game), and not entirely time (as it depends on other variables like field position, score, past game experience, etc.), but it defining the area is an interlinked combination of the two.
About six weeks ago when this bottle episode was created, I came up with another example, and I asked for Joe's opinion on whether it was valid or not. We'll get to what it was in a little but, but first we want to introduce the actual topic of this. We're talking today about computation questions, specifically those in televised competition.
This is the second bottle episode of the summer, and it will slot into the book right after last week's discussion of spelling questions. Unlike spelling questions, which are rare enough to be considered late in your preparation, the question of whether you need someone experienced in computation is often central to your considerations for television quiz bowl. And unlike the circuit, where computation is not considered unless it’s mandated by state authority, it cannot be dismissed as unimportant if present.
Computation is something that we have to discuss, because if it exists in the show’s format, it will take up a lot of questions in those being asked, maybe as high as 15% of the questions you will face. That’s enough to upset the balance in a lot of matches. Because it requires special equipment to visualize the question, or special rules written to account for circumstances, the computation question announces its presence in the rules, giving you ample warning of it being a part of the show. (There is a small possibility that computation may be included in the rules, but not in the rounds you play on. That is because if computation is added to a competition, rules must immediately be drafted for the situation, but the reverse is not necessarily true, special cases can lurk in the rules for years after the situations that require them are removed from the format.)
When I say that computation requires special rules, consider that there are any number of phrasings which could be acceptable or could not be acceptable for an answer that is a fraction or an root of an integer.
- If the problem asks for 3 minus 1 1/2, is 3/2 an acceptable answer? is 1 1/2? is 9/6? is 1.5?
- If the problem asks for the length of a diagonal of a square of side length 1/2, is root 2 over 2 an acceptable answer? is one over radical 2? must it be square root, could it be one half to the one-half power?
Just seeing these examples, you see all the problems that even slightly complex answers could produce for the judges of a show. So if there are computation questions, either the judges have taken these things into account and included them in the rules, or if it’s a new show, they will be forced to take them into account. However it’s more likely that the writers and editors have been burned by this ambiguity previously, and write questions to avoid the situation.
In general this writing to prevent ambiguity serves as a strong check you can apply to answers in televised competition, but for computation it’s especially valuable. If you divide 246 by 6 and get a non-integer result, you’ve probably made a mistake somewhere and should redo your work, even if the delay costs you the point.
What I said about spelling questions applies here as well, if you're asking for an answer to be produced in a shorter time than a national competition dedicated to that particular skill, the scope of what you can ask is highly limited.
When questions can be presented visually to the teams, computation questions can be presented much more compactly to the team facing the question. Instead of describing "a right triangle with one leg of length 8 and a hypotenuse of 17," which takes over five seconds to say, the picture of that triangle can be given in parallel with the instruction "find the length of the unknown leg of the triangle." That ability to give information in parallel moves the question forward faster, and makes it more palatable for the production team (who want as many questions as possible in the time allotted,) and the teams competing (who want as many clues as possible delivered in questions they face.)
To compensate for the limited time, the questions for television will necessarily involve fewer steps to arrive at the answer. You're going to have to work through no more than three steps in your logic to arrive at an answer. And that has some limitations to it. For example, actual mathematical division is going to be more than three steps for anything except those which would have been memorized as the reverse of the multiplication tables. If you're asked to divide, you're looking at a step where you're doing division at each digit, but if you attack the problem as being memorized factorizations you can knock out the problem faster, in two steps (break down the numbers into factors, and cancel out the common factors.)
It was at this point that I pondered the nature of what is given in Lynyrd Skynyrd's "Gimme Three Steps." It's not exactly space, (you might be cover a large amount of distance with each step), or time (you might be able to get up to speed in three steps to give yourself more steps and get out of range)m but the combination of the two (and other parameters like your opponent's ability to get up to speed, and their reaction time), but the output of good outcomes is a combination of the two. (I know, I know, the analogy is forced, but I’ve just mowed the lawn and I’m dehydrated. Bear with me.)
What we don’t consider is if one of those three steps in the song is to throw a chair in the guy’s path, or to dive headfirst through the window. Something that could be accomplished in a step, but is probably considered bending the rules or finding a shortcut. In the song it’s a barfight so everything’s fair game. In televised quiz bowl, these shortcuts are left in by the writer and editor. Below I’ve collected the ways of getting out of television computation faster. These will require a little memorization in advance, so they’re not ideal for a week’s preparation, but if you’re facing a television episode in a month which you know has computation, it’s like getting into a barfight unarmed.
Cancelling terms - Division
(substituting factors for numbers, memorization you need to do this: prime factors of composite numbers up to 50)
Division when it appears is necessarily going to reduce to a whole number. If the writer is asking someone to do long division in their head on stage, they're asking the wrong question.
Cancelling terms - Roots
(memorization required: Squares to 31^2, cubes to 10^3, and powers of 2 to 1024)
These three are just ways of making cancelling terms faster, if you know that 576 is 24^2, you will recognize that any problem that appears with that number in it is going to involve lots of cancellation through division or taking the root down.
Cancelling terms - Factorials
(remember the definition of factorials, and that division of factorials just cancels terms towards 1 until no terms remain.)
This is an iffy case, but factorial's representation onscreen is one of the easiest things to accomplish, so if there's visual presentation, it's worth doing, but it's probably worth doing last among these items.
The Pythagorean Theorem
(memorize: 3-4-5, 5-12-13, and maybe 8-15-17 and that these are ratios, so multiplication can be applied to all three to get to a solution)
If they are doing computation, this will be used at least once, and usually multiple times during a season.
Area of a triangle, rectangle, trapezoid, box, and pyramid and Ohm's law
(memorize the formulas)
Common area formulas are taught in primary school math classes, and these are simply the ones that can be easily displayed and do not involve pi.
Congruent angles, supplementary and complementary
(remember the definitions and the angles, and how angles which are congruent are marked on a diagram)
Knowing which angles in a diagram are congruent, and which angles require subtraction from 180 and which require subtraction from 90 to solve the problem are the basic components of any angle calculation which can be given in visual presentation. Geometry is one of the main reasons that questions are visually displayed, and the information in the drawing is important.
Simultaneous equations
Remembering that you can either add or subtract one equation from another and eliminate a variable is the secret trick used in most simultaneous equation questions. If it’s not that, the remaining case is usually one equation only having one variable defined, and then the solution is to solve that equation and apply the answer to the other equation.
Divisibility tests
(memorize the rules for the numbers 2 through 6, and 8 through 10)
The only tricky bits here are remembering that the rule for 4 (last two digits are a multiple of 4) and 2 implies the rule for 8 (that the last three digits must be a multiple of eight.) and that 6 is the combination of two and three’s rules. If you are faced with the odd case where you’re asked to identify a multiple of 7 like 693, the problem will usually be more easily solved by adding or subtracting 7 to each number.
Sequences and sums
(memorize 5050 as the sum of the numbers from 1 to 100, and spit it out at the mention of Carl Friedrich Gauss)
These aren’t hard problems, but they usually require observation that the sequence questions (“fill in the next number”) usually work from a constant arithmetic difference between the numbers in the sequence, and that the sum of an arithmetic sequence is (first + last) x (number of items in sequence) / 2
Metric prefixes
(memorize the metric prefixes and the exponent of 10 they refer to. In the book we make note of this earlier, so it’s not really news here.)
A question of this form usually will require you to take the number in front of the prefixed unit and bring it into the base unit by multiplication, either for comparison or for deriving a product of two units.
Taking care to learn these knocks out over 75% of the questions that can be asked in television quiz bowl. If you can get to 75% conversion in a category, that’s pretty good, and a significant advantage over unprepared opponents.