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Unsolved Problems — 36 (or more) problems, which until the past ten years or so have remained unsolved.  Each of the problems is simple enough that it is accessible to non-mathematicians.

2004 Logic Quiz — The Archdiocese of Los Angeles (a division of the Catholic church) conducts an annual event for middle school children (ages 11 through 14) called the Academic Decathlon.  This event consists of a number of tests, including the logic quiz, this one from 2004.

2004 State Logic Quiz — After our school won the Archdiocese of Los Angeles Academic Decathlon, we went on to the State competition, where we took 2nd place in the Logic Quiz, and 2nd place overall.

Reciprocal Polynomial — Find values of a and b for which x4 + ax3 + bx2 + ax + 1 = 0 has at least one solution.

Prime Arithmetic Sequence — Find the least possible value of the largest term in an arithmetic progression of seven distinct primes.

Integration Recurrence — Integrate by parts: to find a recurrence relation.

Tetromino Soup — For each tetromino (I, L, O, S, T) find the probability that it will be the one formed by a lattice random walk.

A large product — 1. Prove that 1/64 < (1/2)(3/4)(5/6)... (2009/2010) < 1/44, and
2. Find the smallest positive integer N such that N! is a multiple of 102009.

Mixture Problems — What is the capacity of this jug?  From a 10-gallon keg of wine, a jugful of liquid is drawn off, and replaced with water twice, resulting in a 50/50 mixture.

How Many Solutions? — 1. For values of a does the equation a3x + 3-x = 3 have a unique solution?
2. For what values of m does sqrt(x-5)=mx+2 have two solutions?

Trig is Fun! — prove tan(x-y) + tan(y-z) + tan(z-x) = tan(x-y) *tan(y-z)*tan(z-x)

Set Sums — Prove: if nine distinct integers have a sum greater than 200, then the largest four of them have a sum is greater than 100.

A Set of Rational Numbers — S is a set of rationals such that 1) 1/2 is an element of S, and 2) If x is an element of S, then both 1/(x+1) is an element of S and x/(x+1) is an element of S.  Prove that S contains all rational numbers in the interval 0<x<1.

Counting — 1. X is a set with n elements. Find the number of triples (A, B, C), where A, B, C are subsets of X, such that A is a subset of B and B is a subset of C. 
2. Let m and n be integers greater than 1. Consider an m*n rectangular grid of points in the plane. Some k of these points are colored red in such a way that no three red points are the vertices of a right-angled triangle, two of whose sides are parallel to the sides of the grid. Determine the greatest possible value of k for any given values of m,n > 1.

Reconstruct the Numbers — 1. When I sum five numbers in every possible pair combination, I get the values: 0,1,2,4,7,8,9,10,11,12.� What are the original 5 numbers?
2. When I sum a different set of five numbers in every possible group of 3, I get the values: 0,3,4,8,9,10,11,12,14,19.� What are the original 5 numbers?
3. Is it possible to find a set of 5 numbers in either case above which results in the sums 1-10?  Find an example or prove it impossible.
4. If the above problem is not possible, what is the longest series of sequential sums you can find? For example, problems 1 and 2 have six and five sequential sums, (7-12) and (8-12) respectively.

Triangle Area — What is the area of triangle whose sides are sqrt(61), sqrt(153), sqrt(388) using the trick in which each side is the hypotenuse of three overlapping right triangles.

Card Trick — Charlie has 7 cards numbered 1, 2, 3, 4, 5, 6 and 7 and randomly deals 3 of them to Alice and 3 to Bob. All three people look at the cards that they hold. Can Alice and Bob communicate with each other, in the presence of Charlie, so that after the communication Alice knows which cards Bob has, and Bob knows which cards Alice has, but, for any card except the one he has, Charlie does not know whether Alice or Bob has it? 

Fractal Function — Investigate F(x), a non-decreasing function for all x in [0 1], such that 2F(x/3)=F(x) and F(x)+F(1-x)=1

I'm Thinking of a Number — 1. Sally is thinking of a 6-digit number. The sum of the digits is 43. And only two of the following three statements about the number are true: (1) it's a square number. (2) it's a cube number, and (3) the number is under 500000. What number was Sally thinking of?
2. This is a variation on Guess the Numbers:
I think of 2 single-digit numbers from 1 to 9.  I tell Peter Griffin their product and Lois Griffin their sum.
Peter: "I don't know the numbers."
Lois: "I don't know the numbers."
Peter: "I don't know the numbers."
Lois: "I don't know the numbers."
Peter: "I don't know the numbers."
Lois: "I don't know the numbers."
Peter: "I don't know the numbers."
Lois: "I don't know the numbers."
Peter: "Now I know the numbers."
What are the numbers?
3. Find three three-digit square numbers that together use each of the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 exactly once.

Polynomial Madness1. The zeros of the polynomial x3-33x2+354x+k are in arithmetic progression.  What is the value of k?
2. f(x) = px5+qx4+rx3+sx2+tx+1.  f(1)=4, f(2)=11, and p,q,r,s and t are integers. Prove f(x)=0 has no integer roots.

Three Circles — (see the diagram) Two semicircles are inscribed in a quarter-circle.  Find their diameters.

Buckets and Springs — 1. Given a 3-, 5-, and 8-quart container, how can 4 quarts be measured?
2. 3 springs; 5 pails; (other constraints); What is the shortest time to fill all pails?

Really, Really Big Numbers — When 44444444 is written in decimal notation, the sum of its digits is A. Let B be the sum of the digits of A. Find the sum of the digits of B. (A, B are written in decimal notation.)
2. What is the largest number that can be obtained as the product of positive integers that add up to 100?

Prime Number Puzzles — 1. How many of the three digit numbers that can be made from all of the the digits 1, 3 and 5 (used only once each) are prime?
2. 12345 can be expressed as the sum of two primes in exactly one way. What is the larger of the two primes whose sum is 12345?
3a. Find all prime numbers p such that 2p+1 is a perfect square.
3b. Find all prime numbers p such that 2p+1 is a perfect cube.
4a. Let n be an integer greater than 6. Prove that if n-1 and n+1 are both prime, then n�(n�+16) is divisible by 720.
4b. Is the converse true? That is, if n�(n�+16) is divisible by 720, then are n-1 and n+1 both prime?
5. Let a be the integer whose base 10 representation consists of 119 ones. Prove that a is not prime.
6. Prove composite: n4+4n, n>1.
7. Find the set of all positive primes that are a factor of two consecutive terms n2+3

A Two Player Game — A web application that plays the following game: I'll hand you two numbers -- all numbers are nonnegative integers.  Then you can reduce one or both of the two numbers by any positive integer amount, but you can't make either number negative, and then you can hand me back the slip of paper.  We take turns reducing one or both of the numbers until one of us reduces both numbers to 0, winning the game.  What is a good strategy for winning the game?

Sums of Consecutive Integers — In what ways can 1,000,000 be expressed as the sum of consecutive positive whole numbers?

Rectangle and Quadrilateral Puzzle — (see the diagram) Prove the given quadrilateral is cyclic, and find its angles.

Fermat Pseudoprime Puzzle — Let m=(4p-1)/3, where p is a prime larger than 3.  Show that 2m-1 = 1 (mod m)

Friendly Sets Puzzle — Answer (and prove) six facts about Friendly Sets, which are defined this way: Let S be the set {0,1,2,3,...,m-1} - all remainders modulo some integer m.  Let A be a subset of S and c some integer S.  Let's define rA(c) to be number of combinations of getting c by summing two elements of A modulo m.  Where a1+a2 and a2+a1 are counted as 2 but a1+a1 is counted as 1.  Let two different subsets, A and B, be called friendly if rA(c)=rB(c) for any cS.

1089 — An investigation of the 1089 phenomenon: Take any three-digit number in which the first and last digits are different.  Reverse the digits to get a new number, so now you have two numbers, and each one is the reverse of the other.  Subtract the smaller from the larger to get a third three-digit number.  (If subtracting gives you a two digit number, then please treat it as a three-digit number whose first digit is zero.)  Now add this number to its reverse.  The result will be 1089.

Powers of 3 and Powers of 2 � prove that for every positive integer k, there exists a positive integer m such that 3m + 5 is divisible by 2k.

Cube and Circles Puzzle — A circle is inscribed in a face of a cube of side a.  Another circle is circumscribed about a neighboring face of the cube. Find the least distance between points of the circles?

Handshakes Puzzle — During a party a fellow may shake hands with anyone except himself and his wife (assume all fellows are male, and that they'll only shake hands with someone once). At the end of the party, the Master asks everyone else in the room how many people they shook hands with, and receives 2n+1 different answers. i) Show that the Master's wife didn't shake hands with the most people. ii) How many hands does the Master shake?

Domino Tiling Puzzle — Suppose we use dominoes measuring 2�1 to tile an infinite strip of height 2.  In a typical tiling what fraction of the dominoes will be oriented vertically?  Typical can be defined rigorously by considering all possible tilings of a 2�n rectangle and then letting n go to infinity.

Semi-Periodic Sequence Puzzle — Investigation of the chaotic sequence given by the recursion a0 = 1, a1=1/3, an+1 = 2/3*an - an-1 (n ≥ 1).

Guess the Numbers Puzzle — I think of 2 whole numbers between 2 and 99.
I tell Peter Griffin their product and Lois Griffin their sum.
Peter says: "I don't know what the numbers are."
Lois: "I knew you wouldn't know what the numbers were."
Peter: "Now I know what the numbers are."
Lois: "Now I know what the numbers are."
What are the two numbers?

Recurrence Relation Puzzle 1 — Find the value of the sum F1/1+3F2/5+32F3/52+...+3nFn+1/5n+... (Fi are Fibonacci numbers, F1=F2=1)

Squares & Fourths Puzzle — Find solutions (x,y) of x4 + (x+1)4 = y2 + (y+1)2 

Holey Sphere Puzzle — A circular hole was drilled through the center of a sphere. When the length of the hole was measured along its wall, it was found to be six inches long. What is the volume of the part of the sphere that remains after the material is removed from the hole?

Triangle Partition Puzzle — Is it possible to dissect any triangle into a finite number of acute triangles?

Increasing Function 1 — Investigate function f from Z+ to Z+ where Z+ is the set of positive integers, such that f satisfies these two conditions: (1) f(n+1) > f(n); that is, f is strictly increasing, and (2) f(n+f(m)) = f(n)+m+1 

Nondecreasing Integer Sequence Two — Let a1,a2,...,a2005 be a nondecreasing sequence of positive integers, with t defined as t=a2005.  Let bn be the smallest index, m, for which am ≥ n.
In terms of t, what is the smallest possible value of the sum a1+a2+...+a2005+b1+b2+...+bt ?

Increasing Function B — A sequence x[n] of integers satisfies x[1] = 1 and x[n] < x[n + 1] < 2n + 1 for all positive integers n.  Prove that for every integer k, one can find "a" and "b" such that k = x[b] - x[a].

Integer Function — A sequence x[n] of integers satisfies x[1] = 1 and x[n] < x[n + 1] < 2n + 1 for all positive integers n.  Prove that for every integer k, one can find "a" and "b" such that k = x[b] - x[a].

Locker Puzzle — Students are numbered 1 through n.  The kth student changes the state of every kth locker by opening it if it was closed, or closing it if it was open.  When all n students have paraded past all n lockers, which lockers will remain open?

Freeway Puzzle — Alice was driving along a highway recently for one hour at a constant and very special speed:  The number of cars that Alice passed was the same as the number of cars that passed Alice.  Was Alice's speed the mean or the median of the speeds of the cars on the road?

Mirrors and Light Puzzle — A question about a complicated mirror with light bouncing back and forth.

Two Circles and a Sphere — Two parallel planes are 4 units apart. One plane contains circle A, with radius 2. The other plane contains circle B, with radius 4. If both circles lie on the surface of a sphere, what is the radius of the sphere?

Triangle & 3 Circles — An 1803 Sangaku problem, in which a small circle and isosceles triangle lined up on the diameter of a larger circle with another circle wedged between them.  Prove two particular lines are perpendicular.

Square Inscribed in Sector — You are given a circle of radius 5 with center O.  A and B lie on the circle.  The length of arc AB is 6.  A square with side length x is inscribed in sector AOB such that one corner of the square is on OA, one corner on OB, and two corners on arc AB.  What is the area of the square (and therefore the value of x)?

Rooks Puzzle — A peaceful rook moves on a chessboard like a rook, except that when he comes to another piece, he stops just in front of it (instead of capturing it). Moreover, such rooks always move as far as they can along any row and column (until blocked; if they do not run into a blocker, they fall off the end of the board).  Starting from the configuration of 7 rooks on a 7x7 board shown in a diagram, find a sequence of moves that moves the rook in the upper left (shown in red) to the empty square in the center.

Sum of Series — prove

n n(n-1) n(n-1)...(1)
———  -  —————  + ... + (-1)n-1( ————————— )  =  1/2
(n+1) (n+1)(n+2) (n+1)(n+2)...(2n)

Circles Puzzle — Take a circle and inscribe 4 interior-disjoint congruent circles A, B, C, D that are cyclically tangent (A is tangent to B is tangent to C is tangent to D is tangent to A), and so that each is tangent to the large circle.  A, B, C, and D are colored white in the diagram, right. Then, in each of these four, inscribe four congruent tangent circles in exactly the same way, to get 16 small circles, colored red in the diagram, below. Finally, inscribe a small circle C17, colored yellow in the diagram, in the space around the center of the large circle and tangent to each of A, B, C, D. Which is larger, the yellow C17 or one of the sixteen red circles inside A, B, C, D?

Pirate Puzzle — A question about pirates who distributed some golden coins among themselves, such that:

Cut the carpet — You are given two carpets of dimensions 8-by-8 and 6-by-1. Your task is to make a carpet of dimensions 7 by 10. You are only allowed to cut the 8-by-8 carpet into two pieces. How do you make the cut so that the three carpet pieces can be put together to form the required carpet?

Sides of Square Tiling — This figure is composed of a collection of distinct-integer-sided squares arranged to form a rectangle. What are the rectangle's dimensions?

Prove Composite n^4+4 — For integers n>1, n4+ 4 is not a prime

Arithmetic Sequence of Squares — What's the Longest Arithmetic Progression of Perfect Squares?

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Number Theory answers quite a few puzzling questions.

Sum of SQRTs asks if a<=1, a+b<=5, a+b+c<=14, a+b+c+d<=30 then prove sqrt(a)+sqrt(b)+sqrt(c)+sqrt(d)<=10.

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