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
nonmathematicians.
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 x^{4} + ax^{3} + bx^{2}
+ 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 10gallon 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 a3^{x} + 3^{x} = 3
have a unique solution?
2. For what values of m does sqrt(x5)=mx+2 have two solutions?
Trig is Fun!
— prove tan(xy) + tan(yz) + tan(zx) = tan(xy) *tan(yz)*tan(zx)
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 rightangled 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 110? 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, (712) and (812) 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 nondecreasing function for all x in [0 1], such that
2F(x/3)=F(x) and F(x)+F(1x)=1
I'm Thinking of a Number
— 1. Sally is thinking of a 6digit 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 singledigit 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 threedigit square numbers that together use each of the
digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 exactly once.
Polynomial Madness
— 1. The zeros of the polynomial x^{3}33x^{2}+354x+k
are in arithmetic progression. What is the value of k?
2. f(x) = px^{5}+qx^{4}+rx^{3}+sx^{2}+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 quartercircle.
Find their diameters.
Buckets and Springs
— 1. Given a 3, 5, and 8quart 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 4444^{4444} 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 n1 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 n1 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: n^{4}+4^{n},
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=(4^{p}1)/3, where p is a prime larger than 3. Show
that 2^{m1} = 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,...,m1}  all remainders modulo some integer
m. Let A be a subset of S and c some integer �S.
Let's define r_{A(c)} to be number of combinations of getting c by
summing two elements of A modulo m. Where a_{1}+a_{2}
and a_{2}+a_{1} are counted as 2 but a_{1}+a_{1}
is counted as 1. Let two different subsets, A and B, be called
friendly if r_{A(c)}=r_{B(c)}
for any c�S.
1089 — An investigation of the 1089
phenomenon: Take any threedigit 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 threedigit number. (If
subtracting gives you a two digit number, then please treat it as a
threedigit 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 3^{m} + 5 is divisible by 2^{k}.
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.
SemiPeriodic Sequence Puzzle
— Investigation of the chaotic sequence given by the recursion a_{0}
= 1, a_{1}=1/3, a_{n+1} = 2/3*a_{n}  a_{n1}
(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 F_{1}/1+3F_{2}/5+3^{2}F_{3}/5^{2}+...+3^{n}F_{n+1}/5^{n}+...
(Fi are Fibonacci numbers, F_{1}=F_{2}=1)
Squares & Fourths Puzzle
— Find solutions (x,y) of x^{4} + (x+1)^{4} = y^{2} + (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 a_{1},a_{2},...,a_{2005} be a nondecreasing
sequence of positive integers, with t defined as t=a_{2005}.
Let b_{n} be the smallest index, m, for which a_{m} ≥ n.
In terms of t, what is the smallest possible value of the sum a_{1}+a_{2}+...+a_{2005}+b_{1}+b_{2}+...+b_{t}
?
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 k^{th}
student changes the state of every k^{th}
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(n1) 

n(n1)...(1) 

——— 
 
————— 
+ ... + (1)^{n1}( 
————————— 
) = 1/2 
(n+1) 

(n+1)(n+2) 

(n+1)(n+2)...(2n) 

Circles Puzzle
— Take a circle and inscribe 4 interiordisjoint 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:
 the share of Pirate 1 plus half the total share of the rest (all
pirates except #1)
 = the share of Pirate 2 plus 1/3 of the share of the rest (all
pirates except #2)
 = ...
 = the share of Pirate N plus 1/(N + 1) of the share of the rest (all
pirates except #N)
Cut the carpet — You are given two
carpets of dimensions 8by8 and 6by1. Your task is to make a carpet of
dimensions 7 by 10. You are only allowed to cut the 8by8 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 distinctintegersided squares
arranged to form a rectangle. What are the rectangle's dimensions?
Prove Composite n^4+4
— For integers n>1, n^{4}+ 4 is not a prime
Arithmetic Sequence of Squares
— What's the Longest Arithmetic Progression of Perfect Squares?