For the Math lovers
Buck Mulligan
Will all the 1/k by 1/(k+1) rectangles, for k>0, fit together inside a
1 X 1 square?
Note that the sum of the areas of all these rectangles is 1.
Unsolved Problem 10:
If n is an integer larger than 1, must there be integers x, y, and z,
such that 4/n=1/x+1/y+1/z?
A number of the form 1/x where x is an integer is called an Egyptian
fraction.
Thus, we want to know if 4/n is always the sum of three Egyptian
fractions, for n>1.
Unsolved Problem 11:
Are there any odd perfect numbers?
A perfect number is a positive integer that is equal to the sum of all
its positive divisors, other than itself.
For example, 28 is perfect because 28=1+2+4+7+14.
Unsolved Problem 12:
Is every tree graceful?
A graph is a set of points (called vertices) and a set of lines (called
edges) joinging these vertices.
A tree is a graph with the property that there is a unique path from
any vertex to any other vertex traveling along the edges.
A graph is said to be graceful if you can number the n vertices with
the integers from 1 to n and then label each edge with the difference
between the numbers at the vertices, in such a way that each edge
receives a different label.
For example, a graceful numbering is shown for the following tree with
9 vertices:
(5) (1)---(4)
/ /
(7)---(3)---(9)---(2)
\ \
(6) (8)
The edge labels are the numbers from 1 to 8.
Unsolved Problem 13:
Is there a point in the plane that is at a rational distance from each
of the four corners of a unit square?
A rational number is a number that is the ratio of two integers.
A unit square is a square of side length 1.
Unsolved Problem 14:
What is the value of 1/1+1/8+1/27+1/64+1/125+...?
The nth term is the reciprocal of n^3.
If 3 is replaced by 2, it is known that the series sums to (pi^2)/6.
If 3 is replaced by 4, it is known that the series sums to (pi^4)/90.
Unsolved Problem 15:
Is every Mersenne number square free?
A Mersenne number is a number of the form (2^p)-1 where p is a prime.
A prime is an integer larger than 1 whose only positive divisors are 1
and itself.
An integer is said to be square free if it is not divisible by a
perfect square, n^2, for n>1.
Unsolved Problem 16:
Does every obtuse triangle admit a periodic orbit for the path of a
billiard ball?
We assume that the billiard ball bounces off each side so that the
angle of incidence equals the angle of reflection. If it hits a vertex,
it rebounds along the reflection of its entry path in the angle
bisector of the angle at that vertex. The orbit (or trajectory) is
periodic, if after a finite number of reflections, it returns to its
starting point.
Unsolved Problem 17:
Is there a set S in the plane such that every set congruent to S
contains exactly one lattice point?
A lattice point is a point with integer coordinates.
Unsolved Problem 18:
Are there distinct positive integers, a, b, c, and, d such that
a^5+b^5=c^5+d^5?
It is known that 1^3+12^3=9^3+10^3 and 133^4+134^4=59^4+158^4, but no
similar relation is known for fifth powers. Other remarkable identities
are 27^5+84^5+110^5+133^5=144^5 and
2682440^4+15365639^4+18796760^4=20615673^4.
Unsolved Problem 20:
Are there infinitely many primes of the form n^2+1?
Unsolved Problem 21:
Is every integer larger than 454 the sum of seven or fewer positive
cubes?
Unsolved Problem 22:
Is there a triangle with integer sides, medians, and area?
A median of a triangle is the line segment joining a vertex and the
midpoint of the opposite side.
Unsolved Problem 23:
How should you locate 13 cities on a spherical planet so that the
minimum distance between any two of them is as large as possible?
Unsolved Problem 24:
Is there always a prime number between any two consecutive squares?
Unsolved Problem 25:
Start with any positive integer. Halve it if it is even; triple it and
add 1 if it is odd. If you keep repeating this procedure, must you
eventually reach the number 1?
For example, starting with the number 6, we get: 6, 3, 10, 5, 16, 8, 4,
2, 1.
Unsolved Problem 26:
Given a simple closed curve in the plane, can we always find four
points on this curve that are the vertices of a square?
Unsolved Problem 27:
Are there integers n and x (with n>7) such that n!=x^2-1?
By n! we mean the product of the integers from 1 to n. It is known that
4!+1=25=5^2, 5!+1=121=11^2, and 7!+1=5041=71^2.
Unsolved Problem 28:
The number 3 can be written as 1^3+1^3+1^3 and also as 4^3+4^3+(-5)^3.
Is there any other way of expressing 3 as the sum of three (positive or
negative) cubes?
Unsolved Problem 29:
Let triangles A and B have edge lengths a1, a2, a3, and b1, b2, b3,
respectively. What is the necessary and sufficient condition on the
variables a1, a2, a3, b1, b2, b3 so that triangle A can fit inside
triangle B?
Unsolved Problem 30:
Is every integer the sum of four cubes?
Here we allow the cubes to be positive, negative, or zero. For example,
84=0^3+41639611^3+(-41531726)^3+(-8241191)^3. It is not known if 148,
for example, is the sum of four cubes.
Unsolved Problem 31:
Is it always possible to have n points in the plane (no 3 on a line; no
4 on a circle), such that for every k (with 0 < k < n), there is a
distance determined by these points that occurs exactly k times?
For example, 4 points determine 6 distances. We want one distance to
occur just once, another distance to occur twice, and a third distance
to occur three times.
So far, configurations have been found for n=2,3,4,...,8.
Unsolved Problem 32:
Can you find three integers x, y, and z, such that (x+y+z)^3=xyz?
Unsolved Problem 33:
Is there a constant, A, such that any set in the plane of area A must
contain the vertices of a triangle with area 1?
Unsolved Problem 34:
Are there only finitely many perfect squares with just two different
nonzero decimal digits?
For example, 38^2=1444, 88^2=7744, 109^2=11881, 173^2=29929,
212^2=44944, 235^2=55225, and 3114^2=9696996.
Unsolved Problem 35:
If n points in the plane are not collinear, must there be one of those
points that lies on at least n/3 of the lines determined by those
points?
Unsolved Problem 36:
Is there any value of n other than 1, 2, and 4, such that n^n+1 is a
prime?