Bases
What is 1610 in Base 3?
121
What is the chance of getting a factor of 2 in one roll of a 6 sided dice?
1/2
A man has 53 socks in his drawer: 21 identical blue, 15 identical black and 17 identical red. The lights are out, and he is completely in the dark. How many socks must he take out to make 100 percent certain he has at least one pair of black socks?
40
This is the number you get when you plug one into the formula from Heawood's Conjecture, so this number is the upper bound on the minimum number of colors needed to color a torus. It is the smallest number which cannot be expressed as the sum of three squares, and this is the number of Millennium Prize Problems. Like three, this number forms a twin prime pair with five. For 100 points, give this number, equal to three plus four.
7
What is 2529 in Base 10?
209
What is the probability of randomly drawing a card of the number 5 from a standard deck of 52 cards?
1/13
Each chest has a message on it, but all the messages are lying.
Which chests have treasure?
Left and right
This is the number of solutions to the three-body problem when one of the bodies is assumed to be much smaller than the others. This is number under the radical when finding the ratio between successively higher Fibonacci numbers. The Abel-Ruffini theorem states that polynomials of this degree cannot be solved, and this number is the second Fermat Prime and the third Sophie Germain Prime. This number is the When arranging this many unique objects in a circle, there are 24 possible ways to do so, while arranging this many objects in a straight line there are 120 ways. For 200 points, identify this number, which is also the cube root of 125 and the square root of 25.
5
What is 1046 in Base 3?
1111
What is the probability to draw two 5s in two draws from a standard deck of 52 cards?
1/221
In a group chat with 10 very mathematical individuals such as yourselves, one person asked "Is everyone able to go to my party?" Following that, 8 people responded "I don't know." If the last person is available, what should he/she say?
Yes
A proof of this theorem by President James A. Garfield involves the calculation of the area of a right trapezoid by different methods. Another common method of proving this theorem is by inscribing a square at an angle inside of a larger square, and calculating area of the square both as a whole and as a sum of the component triangles and smaller square. A special case of the law of cosines. For 300 points name this theorem describing the relationship between the length of the legs and the hypotenuse of a right triangle.
Pythagorean theorem
What is 3528 + 1527 in Base 9?
385
On a three by four grid (as in, there are 12 squares in total), how many ways can we move from one edge to the opposite edge in 7 moves?
35
You are asked to guess an integer between 1 and N inclusive.
Each time you make a guess, you are told either
(a) you are too high
(b) you are too low
(c) you got it!
You are allowed to guess too high twice and too low twice, but if you have a 3rd guess that is too high or a 3rd guess that is too low, you are out.
What is the maximum N for which you are guaranteed to accomplish this?
19
The Green-Tao theorem states that there exist finitely long arithmetic sequences of these numbers, and the function pi of n denotes the nth one of these numbers. The Lucas-Lehmer test identifies whether or not (*) Mersenne numbers have this property, and they can be identified by the Sieve of Erastosthenes. Euclid proved that there were infinitely many of, for 400 points, what numbers, which are only divisible by one and themselves, the only even one of which is two?
Prime numbers
What is 12332116 in Base 4? What is 12332125 in Base 5? What is 12332136 in Base 6?
10203030201
A math test has 4 questions. The topic of each question is randomly and independently chosen from algebra, combinatorics, geometry, and number theory. Given that the math test has at least one algebra question, at least one combinatorics question, and at least one geometry question, what is the probability that this test has at least one number theory question? (Don't even try 1/4 lol)
2/5
This is the famous green eye blue eye problem.
On an island, there are k people who have blue eyes, and the rest of the people have green eyes. At the start of the puzzle, no one on the island ever knows their own eye color. By rule, if a person on the island ever discovers they have blue eyes, that person must leave the island at dawn; anyone not making such a discovery always sleeps until after dawn. On the island, each person knows every other person's eye color, there are no reflective surfaces, and there is no communication of eye color.
At some point, an outsider comes to the island, calls together all the people on the island, and makes the following public announcement: "At least one of you has blue eyes". The outsider, furthermore, is known by all to be truthful, and all know that all know this, and so on: it is common knowledge that they are truthful, and thus it becomes common knowledge that there is at least one islander who has blue eyes. Assuming all persons on the island are completely logical, on which dawn (in terms of k) will all the blue eyed Islanders leave the island?
On the k-th dawn.
This man proved that the area bounded by any smooth convex curve and a secant line cannot be represented as an algebraic function of that secant, a result known as his namesake “theorem about ovals.” Samuel Pepys [“peeps”] posed a namesake “problem” to this man regarding the likelihood of rolling sixes from certain numbers of dice. This man devised a thought experiment examining the different paths a cannonball fired off a mountain would take at varying velocities. The discovery of calculus was hotly disputed between Gottfried Leibniz and this man. For 500 points, name this scientist who apocryphally conceived of his theory of gravity after an apple fell on his head.
Issac Newton