50 Million Math Question Fans Can't Be Wrong! (OR CAN THEY)

[Grams's Go-Go Boots]

I am locking this thread to split it PLEASE HOLD.

http://www.youtube.com/watch?v=AZeKsPqU ... re=related

[Mechanical Beasts]

The \"wisted" this is a 3-dimensional parametric curve. Casus irreducibilis refers
to the situation when the roots of a polynomial of this type are all real but
require the use of complex numbers to solve for. Their roots can be found
with * Cardano's formula, but the fact that the roots tend to not be constructible renders
unsolvable two of the three problems of ancient Greek geometry, including trisecting an
angle. Polynomials of this type have exactly one in
ection point and have no maximum or
minimum. FTP, give this adjective describing things with degree 3.
ANSWER: cubic (prompt on \degree 3" before the giveaway)

You claim that there is no difficulty cliff in this TU, but whenever you go from a topic that is taught in an AP Class to a clue that is a simple as "name this adjective that describes things with degree 3" you have a difficulty cliff. If you don't understand why this is the case it is similar to writing a TU on Louis XIV and going from "This man recognized the King of England by signing the Treaty of Ryswick at the end of the War of the Grand Alliance." to "For 10 points, name this "sun king" of France, the fourteenth of a certain name." Additionally, this TU does in fact contain a non-uniquely identifying clue as many high-order polynomials contain one inflection point and no maximum or minimum (example: f(x)=x^5)

Also would it be possible for a moderator to break off the math discussion from this thread into a seperate thread?

And a wrong clue, because a cubic polynomial has a maximum or minimum if there exists an x such that 3ax^2+2bx+c = 0 but 6ax + 2b <> 0.

http://www.wolframalpha.com/input/?i=Plot[x^3-2x%2B3]

[ether a go-go]

Andrew is confusing local extrema for global extrema, or perhaps the question is not clear enough.

[Mechanical Beasts]

Andrew is confusing local extrema for global extrema, or perhaps the question is not clear enough.

Yeah, that's quite unclear, because all that means is "any polynomial of odd degree," and that's pretty weak.

[Charles Martel]

"Polynomials of this type have one inflection point and no maximum or minimum"
The first part of this applies uniquely to cubics, and the second part applies to any polynomial of odd degree. However, it is the only not uniquely identifying part of a clue in the whole tossup, and it's in a sentence where the first part is certainly uniquely identifying.

The thing about the Germain prime was an excuse to put in "exceeding the double of 11 by 1." There was not that much of a difficulty cliff between the second to last sentence, which said the number was 4!-1, and the last sentence that said it exceeded the double of 11 by 1.

[Mewto55555]

So far, you've failed to demonstrate that IMSANITY 1 Math contained anything other than early clues that sometimes started out too difficult, because only a handful of high schoolers would know those clues. The answer lines were for the most part accessible (a term which means convertable, not the other aspects of a good question that Daniel has tried to infuse into it). I have yet to see a single example of a large difficulty cliff, yet you claim that most of the tossups contained one. The one complaint I haven't seen people making about the math is that clues were misordered, yet you also say that "plenty" contained them. Finally, correctness and uniquely identifying the answer are the two most important aspects of questions, and saying that a difficult lead-in is worse than a blatantly incorrect clue is just wrong. IMSANITY 1 Math was not by any stretch of the imagination bad.

CHALLENGE ACCEPTED!

EDIT: If you don't feel like reading things there's a summary at the very bottom.

Here is every math tu from IMSAnity I could find, I’m far too lazy to look at bonuses. I ranked it as either good, medium, or sub-optimal, feel free to disagree with my ratings (I’m probably a bit generous with my medium, considering the strict standards you held that Pythagorean tossup to up-thread). Also its a PDF so it may be problematic copy-pasting.

Actually going through I found that I needed a distinction between the tossups that were bad and the tossups that were terrible, so I added MEDIUM/SUB-OPTIMAL. There’s a lot of stuff so feel free to skip to my conclusions at the bottom.

The \twisted" this is a 3-dimensional parametric curve. Casus irreducibilis refers
to the situation when the roots of a polynomial of this type are all real but
require the use of complex numbers to solve for. Their roots can be found
with * Cardano's formula, but the fact that the roots tend to not be constructible renders
unsolvable two of the three problems of ancient Greek geometry, including trisecting an
angle. Polynomials of this type have exactly one inflection point and have no maximum or
minimum. FTP, give this adjective describing things with degree 3.

Everything pre-power seems rather hard, but I suppose that may or may not be fine. Omar Khayyam is a pretty cool dude that people know about, he may or may not have made a decent middle clue, but whatever. The big problem is the sudden drop in difficulty to the giveaway. There really are no clues on the easy end of medium; there’s plenty of clues that could have gone there (e.g. off the top of my head, stuff like “their derivative is a quadratic” which actually rewards people with Knowledge of some kind). MEDIUM

This is the smallest number of people needed to guarantee at least a 50 percent
chance that some two will share the same birthday. At the 1900 International
Congress of Mathematicians, David Hilbert set forth this many problems as a
challenge to twentieth century mathematics and Book I of The Elements begins
with this many basic denitions, including \point" and \line". Excluding the
* initial arrangement, there are this many ways to reorder four books on a shelf. Itself a
Germain prime, FTP, name this number which proves that 11 is a Germain prime on account
of it exceeding the double of 11 by 1.

Oh god, what is this tossup! I clearly suck as a mathematician because there is no way I would get this before the second-to-last line, and don’t memorize trivia related to the birthday problem or Hilbert’s Problems. Really, 4!-1 is borderline computational, and then you get to a silly and obfuscated giveaway. Difficulty cliffs + trivial clues (I mean this in the trivia sense, not math sense) make this tossup SUB-OPTIMAL.

The sigma variety of this is the set of measurable events in a probability space.
Ideals and modules are the subject of the commutative type, and vector spaces
are the focus of the linear type. Deriving its name from a 9th century book by
* al-Khwarizmi about \balancing" and \restoration", this word comes from the Arabic for
\the method". Groups, rings, and elds are studied in its abstract type, but in general it is
concerned with the existence and determination of solutions of equations. FTP, name this
branch of mathematics, the bane of many high school freshmen.

I’m skeptical that sigma is the best lead-in, but I’ll let that go. You never give a more specific noun than “type” which should be a red flag that this is not the best answerline. Your giveaway is terrible, and linear comes way too early. SUB-OPTIMAL

Some methods of attacking this problem involve looking for the largest interior
angle or considering the convex hull. Dynamic programming techniques can solve
this problem exactly in exponential time, but the Nearest Neighbor heuristic
can nd an approximate solution to this problem much faster. Cutting-plane
methods to developed by Dantzig, Fulkerson, and Johnson to solve this on *
United State capitals. By a reduction of the Hamiltonian Cycle Problem, Richard Karp
showed that this problem is NP-complete. Finding the cheapest of n factorial possible paths
is, FTP, this problem which seeks the shortest possible route through n cities.

This is in fact not terribly written (to my knowledge, I don’t really know CS), but I’d love to see conversion data, as I’d be very shocked if lots of high schoolers knew what this was. Also, I’m skeptical about the first clue (lots of things are found with convex hulls and interior angles!) If you can convince me its answerable (with staaaats), I’d change it to medium, but for now its MEDIUM/SUB-OPTIMAL.

Riemann's namesake type of this gure is another term for the extended complex
plane, and Kepler's Conjecture concerns the most ecient way to pack these. For
triangles embedded on this surface, the properties of similarity and congruence
coincide, since area is proportional to the angular excess. This gure * maximizes
volume for a given surface area, the latter of which is four times the area of one of its great
circles. The collection of points in 3-dimensional-space equidistant from the center is, FTP,
this higher-dimensional analog of a circle.

Your first sentence seems backwards, why would you not describe the extended complex plane (or instead of using excess jargon, say things like “the complex plane and the point at infinity”) before dropping Riemann; he’s the namesake of a lot of other things too! The “angular excess” clue seems way too obfuscated, why not just say “triangles on this surface have angles summing to more than 180”. Also what in the world does “properties of similarity and congruence coincide” mean? MEDIUM since I guess its accessible and pyramidal.

In 2004, Marcus and Tardos proved a conjecture of Stanley and Wilf on the
number of these avoiding a certain pattern. These can be classied as even or
odd depending on the number of transpositions needed to express them. One
is called a derangement if no object occupies its original position. A random
one can be generated with the Knuth * shue. There are six of these for the letters
ABC, 120 for the numbers one through ve, and 5040 for seven people standing in line.
A reordering of a set of objects is, FTP, this mathematical structure, often paired with
combinations.

I’m not actually sure what to think about this tossup; it strikes me a lot like TSP except much more cliffy and slightly less accessible. As someone who Knows Things about math contests, it mostly looks fine, except the first clue seems a bit weirdly worded, and the even or odd should come after transpositions since as is there are lots of even/odd things. Also, can you really not do better than saying “these” every sentence with no noun following? MEDIUM/SUB-OPTIMAL

The Chebyshev Inequality guarantees that the probability that a random variable
di
ers from its mean by more than alpha is at most this quantity divided by alpha
squared. This quantity preserves addition when applied to independent random
variables. For the outcome of a fair die, this quantity is thirty-ve twelfths, or
about 2:92. Dened as the * expectation of the square of the quantity of a random
variable minus its mean, this parameter is usually calculated as the mean of the square
minus the square of the mean. Usually denoted by sigma squared, FTP, name this measure
of the spread of a probability distribution.

As someone who DOES MATH CONTESTS I can tell you that there is in fact ANOTHER Chebyshev Inequality which is a bazillion times cooler, if not slightly less well-known outside the quizbowl world. If you’re shooting for accessibility, why in the world toss this up instead of standard deviation (and why not mention the square of the standard deviation in the giveaway?) SUP-OPTIMAL

All solutions to a logistic di
erential equation have either one or two of these.
Their name was coined by Appolonius to mean \not meeting", although modern
interpretation allows them to * intersect their corresponding curves. For rational
functions, the quotient resulting from long division yields the horizontal and oblique types
and the zeroes of the denominator help determine the rest. The graph of arctangent of x
has two of these, but the graph of tangent of x has innitely many vertical ones separated
by distances of pi. The limiting behavior of functions near positive or negative innity are,
FTP, these lines that graphs approach.

That first clue strikes me as rather unclear, its possibly fine. I’m not sure how the etymology of the name is at all useful to someone who knows math. The tossup seems to cliff way early (zeroes in the denominator anyone? things that curves kind of almost meet?) Also your giveaway is a tad confusingly worded. SUB-OPTIMAL

In algebra, a field must have at least this many elements. This is the sum of the
reciprocals of the triangular numbers as well as the minimum possible sum of a
positive number and its reciprocal. The * Goldbach Conjecture asserts that this is
the largest even number not expressible as a sum of two primes. The number of subsets of
a set of size n is this number to the nth power. The base of binary arithmetic is, FTP, this
smallest prime number.

Seems like a lot of middle clues are hard (I would imagine the triangular number thing is harder than the field, because not many kids are going to whip out a pencil and do some telescoping sums in the span of six words), and I think the subset thing is harder than Goldbach. Still, I’ll give this the first GOOD (one or two more easy late-middle clues would be even better!)

Roth's Theorem guarantees the existence of these groups of numbers in any
dense subset of the natural numbers, and the Green-Tao Theorem guarantees
arbitrarily long ones in the primes. Dirichlet showed that these groups always
contain prime numbers if the rst two terms are coprime. A young * Gauss
developed a method for summing the elements of one of these, which was to average the rst
and last terms and multiply by the number of terms. Discrete linear growth characterizes,
FTP, this group of numbers formed by iteratively adding a common di
erence to an initial
term.

There’s no need to start a tossup with three hard clues that extend until the power mark. It seems to cliff quickly, and I don’t know why you feel a need to say the word “linear.” MEDIUM/SUB-OPTIMAL

Leonhard Euler left his native Switzerland for this country to join its academy of
sciences. One mathematician from this country rst proved Bertrand's Postulate
that there is always a prime number between any positive integer n and 2n, and
another, along with Janos Bolyai, developed non-Euclidean geometry. In 2006,
the Fields Medal was rejected by the mathematician from this country who
proved the Poincare Conjecture.* Chebyshev, Lobachevsky, and Perelman hailed from,
FTP, this country in which Euler was a member of the St. Petersburg Academy.

What the hell? SUB-OPTIMAL

Only 47 numbers are known to have this property, the largest of which has
nearly 26 million digits. Euler proved that the Mersenne primes are in one-to-
one correspondence with the even types of these. No * odd numbers are known to
have this property, although if any do exist, they have at least nine distinct prime factors.
Numbers with this property include 496 and 8128, although more common examples are 6
and 28. FTP, name this property of numbers whose proper divisors sum to themselves.

This seems to cliff a ton at the power mark, although I’m not sure what clues are good for perfect numbers, and I’m not convinced that the lead-in is good. If anything, maybe say something about how like 496 equals 16 times 31 and move that earlier? I’m also skeptical this answerline is super-convertable, but maybe. I think the biggest problem with this tossup is the answer line so MEDIUM/SUB-OPTIMAL.

This is the only one of Peano's axioms for the natural numbers that is second-
order. A transnite version of this can be used on large innite sets. This exists
in strong and weak forms, both of which are logically equivalent to the well-
ordering principle of the natural numbers. In proofs, it is used by establishing
a * base case and then demonstrating a namesake step invoking a namesake hypothesis.
Often stated as \if S of zero is true and S or n implies S of n + 1, then S of n holds for all
natural numbers n", FTP, name this principle and proof technique which progresses from
specic to general, a complement to deduction.

YES THIS TOSSUP TOTALLY IS A COMPLETELY FORMED SLOPING PYRAMID ALL THE WAY UNTIL THE BOTTOM. NOT! SUB-OPTIMAL.

The Sylvester-Gallai Theorem states that if a nite collection of points in the
plane does not have this property, then some line contains exactly two of them.
The center of the nine-point circle, orthocenter, and circumcenter of a triangle
have this property, and a set of points in the plane is said to be in general position
if no * three have this property. Two vectors have this property if they are multiples of
each other. Any three points not having this property determine a unique plane. Any two
points have, FTP, this property in which a single line can be drawn through them.

This would be really neat if it were accessible, although the vectors clue seems a tad misleading (I’ve always just referred to them as “vectors pointing in the same direction” or “parallel”). Cool idea, pretty well-executed, but I think too hard for most high schoolers. SUB-OPTIMAL

One of these seeks a method to nd generators of rational points on elliptic
curves, and another investigates the complexity gap between computability and
veriability. Two of these have applications in physics, one in
uid mechanics
and the other in quantum eld theory. The most famous of these involves the
distribution of prime numbers. To date, only one of these seven problems. The
solution to any one of these comes with a * $1,000,000 prize from the Clay Mathematics
Institute. First introduced in May 2000, FTP, give the collective name of these problems,
including the Poincare Conjecture, P versus NP, and the Riemann Hypothesis.

I’m loathe to use the word transparent, but I think I have to here. Also, how accessible is this really? SUB-OPTIMAL

This mathematician generalized the binomial theorem to powers of multinomials.
His unpublished proof of Fermat's Little Theorem is the rst known to exist. He
made two trips to England in the 1670's, which would later become a source of
contention. Among his most well-known contributions are notational, including
the * elongated \S" and \d y over d x". These appeared in his book New Method for Maxima
and Minima, and Also for Tangents, Which is Not Obstructed by Irrational Quantities, in
which he laid out his most important discoveries. FTP, name this German mathematician
who, independently of Newton, developed calculus.

This strikes me as too hard for high school, at the very least until the last line (isn’t that all nearly everyone knows about Liebniz?) The top of this pyramid seems a bit hard, and I think it falls very much prey to the earlier mentioned “gettable but only at the very very end”. SUB-OPTIMAL

The x-coordinate of this point can be computed as the double integral of x \d
A" divided by the area of the gure, and the y-coordinate satises the analogous
formula. Pappus's theorem says that the volume of a solid of revolution is
the area of the revolved gure times the distance traveled by this point, which
remains xed under any isometries of the gure. For a * triangle, its coordinates
are the arithmetic means of those of its vertices, and this point is the intersection of the
triangle's medians. Also called the barycenter, FTP, identify this point, the center of mass
of a gure.

This seems a tad hard, why not make it on “center of mass” and work in physics clues. Pappus’ Theorem is harder than the line before it. MEDIUM/SUB-OPTIMAL

In the graphs of this type of function, a horizontal translation is equivalent to
a vertical dilation, as this type of function satises f of the quantity x plus y
close quantity equals f of x times f of y. An equiangular spiral is the result of
graphing this kind of function in the polar plane. The product of any two of
these functions is * also of this type, as is any power of one of them. Di
erentiating one
of these functions yields a multiple of that function. The inverses of logarithmic functions
are, FTP, this type, subject of a namesake growth.

This is a neat idea, I like it. The early clues seem fine, but I’m skeptical the f(x+y)=f(x)f(y) is unique (I’m pretty sure a substitution takes it to the Cauchy functional equation,so it probably has some similarly crazy solutions, but this problem can be eliminated by just adding the word “continuous”). The next few clues seem opaque, why not mention more about exponential growth (e.g. its differential equation), and to make the giveaway easier you could definitely say the word “power.” Still, I’ll give this a GOOD.

This unary operation can be dened in any quadratic extension, and in its most
common form, it forms the only nonidentity ring isomorphism of the complex
numbers xing the reals. As a result, for every * root of a polynomial with real
coefficients, this is also a root. Most often denoted with an overline, it is used when dividing
as it rationalizes the denominator. Found in its complex form by negating the imaginary
part, FTP, name this number given by the formula a minus b i.

The top half seems rather confusingly worded (maybe add a matrixy clue?), and the tossup itself has a tough answerline. The giveaway (and for that matter the middle to end) could definitely be clarified (the complex conjugate is not a-bi, the complex conjugate of a+bi is a-bi, and it’d be nice if that distinction were made at least once). Also, that polynomial root clue is sketchy, I know what you mean but its way too hard to parse on the fly. SUB-OPTIMAL

The set of numbers which can be expressed as a sum of two of these is closed
under multiplication according to a formula of Brahmagupta and Fibonacci.
These integers have an odd number of factors, and every natural number is the
sum of four of them by Lagrange's Theorem. The sum of any two consecutive *
triangular numbers is one, as is the sum of any initial segment of the positive odd integers.
When divided by four, they must leave a remainder of 0 or 1, and they cannot end with the
digits 2, 3, 7, or 8. FTP, name this group of integers, obtained by multiplying an integer by
itself.

The odd number factors bit is probably more well-known than a bunch of stuff that comes after it, and there’s not that much easy except for the giveaway. This tossup has a lot too many medium-difficulty clues. MEDIUM/SUB-OPTIMAL

A theorem of Fortuin, Kasteleyn, and Ginibre states that increasing events are
positively correlated and is known as the FKG one of these. Jensen's states that
convex functions lie below their secant lines, and Markov's shows that a random
variable cannot be large with high probability. A more notable one implies states
that * x plus y is at least 2 times the square root of x y and is known as the arithmetic
mean-geometric mean one, and another states that the length of one side of a triangle is
at least the di
erence and at most the sum of the other two side lengths. Relationships
comparing two quantities are, FTP, these, exemplied by e to the x is greater than or equal
to one plus x.

This is an interesting idea, but not very well-executed (or possible to execute?) There’s no noun to describe it which is problematic, and the early clues are way too hard (it’d be much better if it were all triangle, am-gm, and maybe something else). Also, the word “at least” is really early. SUB-OPTIMAL

A xed point of a function is an intersection of its graph with this entity, which
is the best linear approximation to y equals sine of x around x equals zero.
Multiplying the graph of a function by the matrix with rst row 0 1 and second
row 1 0 re
ects across it, which produces the * graph of the function's inverse, and a
function composed with its inverse yields this curve. Having one intercept and making a 45
degree angle with the positive x axis, FTP, name this line with a slope of one and passing
through the origin.

This is actually a really cool and mostly well-executed idea (perhaps a bit hard for a while though)? I like it! GOOD.

While this curve is not sinusoidal, its height is proportional to its arclength,
causing it to obey simple harmonic motion, and Christiaan Huygens used this
isochronous curve to design a more accurate pendulum clock. As shown by
Newton and Bernoulli, it is also a solution to the * brachistochrone problem, as a
frictionless object accelerated only by gravity will travel between its endpoints faster than
along any other curve. Described with the parametric equations x equals r t minus r sine t
and y equals r minus r cosine t, FTP, identify this curve swept out by a point on a circle
rolling along a line.

I’m not sure how gettable this is (ANSWER: not very!) by most high schoolers, and you keep saying the same thing over and over again (a ball rolling down this takes the same amount of time! isochronous! brachistochrone!) SUB-OPTIMAL

Powers of this number, equal to twice the cosine of =5 radians, were used
by John Conway to show that in the game of Desert Solitaire, no armies can
advance more than ve units. Contrary to popular belief, some historians now
believe that this number, as opposed to the square root of two, to be the rst
number ever proved to be irrational. Research has discredited theories about
the * aesthetic value of this number, which can be obtained as the limit of the quotient of
consecutive Fibonacci numbers. FTP, name this number which exceeds its reciprocal by 1.

Not sure how useful the Desert Solitaire or history clue is, I don’t think both are needed along with the cos(pi/5). This tossup doesn’t have very much math, even though there’s a lot of stuff about it, and the giveaway is not really much of a giveaway at all! SUB-OPTIMAL

This country is home to the Miklos Schweitzer Competition, a 10-day mathematics
contest for undergraduates, which is sponsored by the Janos Bolyai society.
With the exception of the USSR and China, this country has won the International
Mathematics Olympiad more than any other. This country's eponymous
algorithm was developed by Kuhn to solve the assignment problem. One mathematician
from this country worked with Morganstern to write * Theory of Games
and Economic Behavior. In addition to John von Neumann, this country is home to a mathematician
with a namesake distance measuring collaboration. FTP, name this European
country, the home of Paul Erd}os.

And I thought MSHSAA math was bad! SUPER DUPER SUB-OPTIMAL!

It is widely believed that the cuneiform writing on the Plimpton 322 tablet
represents these. Babylonians had a formula to compute these, although it was
only a special case of the general formula given in Book 10 of Euclid's The
Elements. Rational points on the unit circle are in one-to-one correspondence
with these * groups of numbers. Every positive integer appears as an element in at least
one of these, but in the primitive type, in which the elements share no common factors, only
odd integers can be the largest element of the three. Including 5 12 13 and 3 4 5, FTP, name
these integer solutions to x squared plus y squared equals z squared.

This is not too great, the entire first half is basically unbuzzable. Also you are wrong, I CHALLENGE YOU TO TELL ME A PYTHAGOREAN TRIPLE WHICH CONTAINS THE NUMBER 1 OR 2. Nifty answerline, but question needs some work. SUB-OPTIMAL

In ring theory, this number tells how many times the multiplicative identity must
be added to itself to obtain the additive identity. Euler's one of these is equal
to 2 for the plane. Setting this type of polynomial equal to 0 yields this type
of equation, and solving that equation yields the * eigenvalues of a matrix and thus
solutions to linear di
erential equations. This word can serve in place of \indicator" when
describing a function which is equal to 1 at every element of a set and 0 everywhere else.
FTP, give this word which composes, along with the mantissa, the common logarithm.

This giveaway is not too giveaway-y (in fact the easiest thing in this tossup is the matrix polynomial!), and the answerline is likely too hard to write on. SUB-OPTIMAL

This is the maximum value of the minimum degree of a nite planar simple
graph and the smallest number of vertices in a non-planar graph. This number
is the smallest degree of a polynomial which cannot be solved in radicals. In
geometry, this is the number of faces meeting at each vertex in the * icosahedron,
and hence, by duality, is the number of sides on each face of the dodecahedron. FTP, identify
this number, the number of Platonic solids.

This is mostly good, especially the first half. Easier easy clues need to be at the end though. MEDIUM.

Bessel's correction is used to make this kind of variance an unbiased estimator.
This can be e
ectively done in clusters or in strata, but the convenience method
is frowned upon. Its namesake * proportion is the best point estimate of the success
probability. The e
ect of its namesake size on its namesake mean is the subject of the Law of
Large Numbers, and its namesake space consists of all possible measured events. Referring
to either a representative subgroup of a population or to data from that subgroup, FTP,
identify this term for the collecting of information for statistical analysis.

I can’t tell what to make of this tossup, but none of the judgements I’m drawing are very positive. Cluster/strata comes way early, and I don’t think anything after that is much easier, so difficulty sort of plateaus for about 4 lines. SUB-OPTIMAL

In the 1970s, Thurston proved that each of these is either hyperbolic or is of
the torus or satellite types. The chirality of these objects can be used to study
that of certain molecules. Two of these objects are equivalent if one can be
transformed into the other through a sequence of Reidemeister moves. They
can be distinguished by their associated polynomials, such as the Alexander,
Jones, and HOMFLY, or by the * Gordian distance between them. FTP, name these
closed one-dimensional surfaces embedded in three-dimensional space, popular examples of
which are the pretzel, trefoil, and gure-eight.

This is way too hard; no one’s getting it before pretzel/trefoil unless they either know non-zero amounts of stuff about know theory (that’s knot too many people!) or can figure it out at Gordian. SUB-OPTIMAL.


Summary:

Upon tallying my results, I found:

3 good
3 medium
6 medium/sub-optimal
17 sub-optimal
1 SUPER DUPER SUB-OPTIMAL

or 10% good, 10% medium, 20% medium/sub-optimal and 60% sub-optimal. This is not great.

[Wackford Squeers]

I don't buy the "number of years in a classroom argument." The vast majority of time spent in math classes in American high schools is spent learning how to solve math problems. The amount of time spent discussing math theory and mathematicians could probably be condensed into a single semester. Arguing that math concepts learned to help solve problems can be easily translated into buzzable, accessible clues is comparable to arguing for Literature questions on "characterization" or "proof-reading". I'll grant that IMSANITY II was a major improvement on its predecessor, but neither set truly convinced me that quiz bowl would come closer to reflecting some sort of ideal of a balanced education by upping the math distro. We have to recognize that Quiz Bowl can never reward all forms of knowledge equally. It has a clear and acceptable bias towards what is basically "book-knowledge." The fact that people can't get points for much they've learned studying for math contests or improving their creative writing is an unavoidable and eminently tolerable fact.

[Scaled Flowerpiercer]

Going through Max's analysis of the IMSANITY I tossups, I agree with his analysis for the most part, and notably as far as "my degree of math knowledge" I could power most of them, and could answer every one not about cycloids. I feel that what these questions in particular suffered from are difficulty cliffs and sometimes plateaus, and I think that the former of these is something that can be hard to avoid at times and the latter is something which is very improvable.

For one thing, sometimes it is hard to order math clues, as many of the harder than high school level clues can be difficult to judge - for one thing, I personally think that Max's comment on the centroid question that Pappus was harder than the integral formula for calculated it was wrong, as I think that Pappus is easier, if only because anyone who has learned the Theorem of Pappus would be able to buzz instantly after some "distance traveled by the ___ clue" with it, whereas the integral clue made me think Center of Gravity for a moment, due to the similarity of CoG and Centroid formulae. And many of the "plateaus" were just the result of bad clue ordering - for example, the perfect square question, odd number of factors was easier than the next three lines, easily, and re-ording would restore some degree of a pyramid. Also, while I am commenting on Max's comments, I personally thought that dy/dx ==> Leibniz was a very easy clue, but I guess that the fact that I think that this, which is a clue based on knowledge of calculus, is the easiest non-giveaway thing about Leibniz in itself proves it is too difficult

Now, as for difficulty cliffs, I think that - especially with high-school accessible answer lines - this can be a great problem, as tossups will almost always use as beginning clues facts which are inaccessible to those whose math knowledge is at a "normal" high school level, and there will therefore be a cliff once clues legitimately contained within a HS level curriculum is reached.

Also, another issue I found with some of these tossups, that I could see as being a fairly common issue with Math Tossups, is the use of a specific version of a formula which has been written in different ways, for example, in the variance tossup, describing Tchebycheff's inequality using "the amount by which the RV differs by alpha," "alpha" is not only not really necessary, but to some small degree confusion to someone like me, who has only seen the formula written with the parameter epsilon ... of course, this tossup could easily be written without naming this constant.

A final note I have to make, is that it seems from this game that "meta math tossups," such as asking about the Millenium Prize Problems, or country answerlines, seem like they tend to be awful...math tossups should stay as math tossups and not try to be about anything else.

PS: a lot of the questions copy/pasted rather horribly, which led to a couple of times when I found a question difficult to answer in Max's post due to the presence of the word "gure," somehow created from "figure"

[nadph]

Perhaps I am being idiotic, but I unsure of what exactly Max's detailed critique proves, other than that IMSANITY math was of highly variable quality, and consequently its writers need more experience before their writing can be said to be good. This is also true given that we have at least two other (mostly) pyramidal sets with a significant (2/0 out of a 20/0 round) distribution of math tossups, namely the past two iterations of Scobol Solo; those math tossups certainly don't seem utterly terrible at a cursory read-through. (Someone who actually played Scobol Solo can correct me here.) Is there something (other than the lack of bonuses) that makes these sets ineligible for discussion? I haven't seen them mentioned so far.

[Wilberbeast]

In my opinion, there should be two main criteria when considering whether or not to include a category in quiz bowl.

1. Is it relevant in academia: The effective of purpose of quiz bowl, in my view, is to further academia, by pushing students to learn beyond that of what they are expected to in basic-level courses, in what happens to be a fun way.
2. Is it implementable? This is important for obvious reasons.


So is math relevant in academia? I think so. With the vast majority of academic institutions having a basic requirement of mathematics in order to graduate, the position of math in academia is undeniably validated.
The main questions seems to be whether or not it is implementable. The argument that some tossups written on the topic were non-ideal doesn't really say anything. As such, the one-by-one analysis of mathematics tossups and bonuses in any given set is fairly negligible in the larger picture. So then how do we prove whether math is implementable or not? In my own experience, I find it to be quite implementable by the mere reason that any criticism of a math tossup can generally be easily fixed. Some examples:

There are not enough answer lines that are askable.

There are many, many subcategories within mathematics. Algebra, Geometry, Trigonometry, Calculus, Noneuclidean geometry, graph theory, number theory, linear algebra, and set theory are just a few I can name off of the top of my head. Yes, the average chump may not run into these in their average class, but that is exactly the purpose of Quiz Bowl: to push academia beyond that of the average classroom. Whether the tossup regards a number, a theorem, a mathematician, a famous problem, or some sort of mathematical construct (of which there are many), there is a lot to be asked about.

It is too difficult to write math tossups.

Honestly, I think this is merely the argument of those who do not know enough about "obscure" math topics to write about them properly. Pointing out the faults of specific tossups does not prove this argument, either.

There is no room in the distribution.

The fact that there is often times more Trash or Pop Culture (ie non-academia) in a set than Mathematics is not very helpful to this argument. Even a mere 1/1 distribution would not be difficult to substitute into a set, as has been shown by precedent.

[Kenneth Widmerpool]

Perhaps I am being idiotic, but I unsure of what exactly Max's detailed critique proves, other than that IMSANITY math was of highly variable quality, and consequently its writers need more experience before their writing can be said to be good. This is also true given that we have at least two other (mostly) pyramidal sets with a significant (2/0 out of a 20/0 round) distribution of math tossups, namely the past two iterations of Scobol Solo; those math tossups certainly don't seem utterly terrible at a cursory read-through. (Someone who actually played Scobol Solo can correct me here.) Is there something (other than the lack of bonuses) that makes these sets ineligible for discussion? I haven't seen them mentioned so far.

I think that Max's critique proves that insofar as IMSANITY was supposed to be a model of a good set that paved the way for a new quizbowl with increased math content, it failed. I haven't seen Scobol Solo, but maybe if it were scrutinized about as heavily, it could succeed (or maybe it couldn't).

[Cheynem]

I kind of lost track of what the argument is.

Good math questions should be encouraged. Tournaments should feature math questions within the science distribution. Good writers and editors, especially those with math knowledge, will (hopefully) produce good math questions. Tournaments with less capable math writers might feature somewhat less math. Tournaments that with more capable math writers might have a little bit more math. People with real math knowledge should provide input if they feel like their knowledge isn't being rewarded. I'm skeptical that a tournament needs to have a ton of math, but accessible math questions should probably appear more often in high school sets than some other topics. I would hope no one would disagree with these statements.

The IMSANITY argument doesn't really interest me that much. From what I can gather, the writers were mostly inexperienced, so no matter their math expertise, there was probably bound to be some bad ideas or problematic questions. What is more important is thinking about how math questions in general (whether by NAQT, HSAPQ, or housewrites made by people with less math backgrounds than the IMSANITY or LIST writers) can be improved. In this light, it might be more productive to try to discuss what makes ideal clues or answers for math questions rather than go back and forth about one particular tournament's quality (I'm not trying to curb discussion here--you can keep doing whatever you want, I'm just trying to get something productive out of this thread).

[Smuttynose Island]

This argument began with my contention that a good question writer or editor with average math knowledge would be able to produce better math questions than a bad question writer or editor with a lot of math knowledge.The reason why Max's critique of IMSANITY's math TUs is pertinent is because Adam claimed that IMSANITY's math supported the opposite of my claim, namely that people knowledgeable in math would, regardless of their competency at actually writing questions or being good editors, produce, on average, better math questions. It was also made clear and, I believe agreed upon, that a person who is both knowledgeable at math and a good writer or editor would, on average, produce the best mathematics questions. This distinction was important because it determined, by extension, who we felt was qualified to judge whether or not a set's math was "good." Adam, in an attempt to demonstrate that you could write lots.of math questions in a set week, cited the math in List II andIMSANITY II, but failed to offer evidence that supported those .claims outside of "these people who know math think that the math in this set is good,"I which I contended was invalid evidence.

This was never a debate, or atleast I never made the claim that, math questions are neither important nor that experimenting with expanding the math distribution is wrong.

[Eccles cake]

There was a good discussion about mathematics in the ACF distribution topic starting here if you guys want to read through it. A lot of the points made there also apply to high school, and I won't rehash them here.

Anyway, the conclusion we came to in that discussion was that math for the purposes of college shouldn't have to enforce mandate a .5/.5 distribution because if math was underrepresented then the invisible hand of quizbowl distribution would naturally cause better math questions to be submitted to packet submission tournaments and the quizbowl distribution would hence evolve more organically. However in the case of high school where packet submission is non-existent, having a top-down institution of a higher math distribution becomes necessary.

The dearth of quality math questions is less a factor of increased math distribution being unsuitable for quizbowl and more a factor of bad question writing (something Nikhil brought up). The existence of bad math questions should not be used as a reason for not allowing for the math to increase unless their existence shows that good math questions are impossible. This is something which can be shown to be false since NAQT and others have written perfectly fine math questions. What we should be looking at are the answer lines for math tossups and whether the answer lines can produce good, accessible questions.

To address one other point, I don't think it's productive to go on and on about how many years of mathematics people take in high school. Mathematics is an undeniably important field of study. This should be a given. Thus, whether or not mathematics should deserve a higher distribution should be based only on things that affect a packet or tournament's playability.

To sum up, for a subject to deserve a higher distribution, I believe it needs to satisfy three conditions (this only applies to high school since packet submissions should correct for this in the college game): 1) if expanded, there must be a large enough set of possible, good answer spaces for questions; 2) the questions must be accessible; 3) the current distribution is unfair to the subject in proportion to its real-life academic importance.(*)

If nothing else, IMSANITY has shown that condition 1 is possible. The answer lines selected above were good, even if the questions themselves may have been subpar. Condition 2 also holds so long as the expansion of math questions expands more to discrete math, algebra, geometry, etc. as opposed to higher leveled maths which I think is certainly possible and is being done. Condition 3, in my opinion, is undeniable. The realworld academic significance of mathematics is huge, and the distribution of mathematics within quizbowl is miniscule in comparison. To have sets which only contain 3/3 mathematics seems to be an oversight considering its academic significance.

Anyway, I don't know the degree to which the mathematics distribution should be increased, but I think I put forward a decent argument to its increase.


(*) As an aside, I think it might be helpful in quizbowl discussions to axiomatize certain things about the game (that is, produce concrete assumptions and conditions from which we can base inferences), which I attempted to do here. If anyone wants to provide a different set of conditions for increasing quizbowl distribution of certain subjects (or even conditions for other things), that might be a better route of discussion to go than to discussion countless examples ad nauseum.


EDIT: I just read the change in distribution for NASAT which is great news and which I hope will lead to a permanent change in the science distribution in quizbowl.

[Mewto55555]

The reason why Max's critique of IMSANITY's math TUs is pertinent is because Adam claimed that IMSANITY's math supported the opposite of my claim, namely that people knowledgeable in math would, regardless of their competency at actually writing questions or being good editors, produce, on average, better math questions.

Daniel hit the nail on the head here; I strongly disagreed with the statement "people who are experienced with math but not with quizbowl writing are better at writing math than those who are good at writing but not-experienced with math," as well as the idea that IMSAnity 1 should be held up as the paragon of math question-writing. Also, the reason LIST II has met with approval from the mirrors so far is not because I can solve some problems on the USAMO; its because LIST II was the third tournament I've head-edited in the past two years. If you ask any MOPper to write some quizbowl questions, I can guarantee you the only other one who would produce excellent questions would be Anderson Wang (who notably does quizbowl and has writing experience).

That's not to say I don't think there should be math; a lot of the problems in the IMSAnity set could have been easily avoided and the distribution could have still been filled. Math is definitely writable; I'd hazard a guess you could probably hit at least 2/2 if not 3/3 of accessible well-written math in a set at regular difficulty, but that certainly doesn't mean we should have that much!