Tuesday, August 14, 2012

Mathematics for the Liberal Arts

This problem is from Mathematics for the Liberal Arts'' (Wiley, 2012).
Leonhard Euler (1707-1783) proved the formula ${1\over 1^2} + {1\over 2^2} + {1\over 3^2} + \cdots =\frac{\pi^2}{6}.$ Can you use Euler's formula to find the exact value of $\frac{1}{1^2}-\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+\frac{1}{5^2}-\frac{1}{6^2}+\cdots ?$

Sunday, August 12, 2012

Six Mathematical Puzzles

Let $S = 1 + 11 + 111 + \cdots + 1\ldots 1$, where the last summand has $111$ decimal digits each equal to $1$. What is the sum of the digits of $S$?

Suppose that five particles travel back and forth on the unit interval $[0,1]$. At the start, all five particles move to the right with the same velocity. When a particle reaches $0$ or $1$, it reverses direction but maintains its speed. When two particles collide, they reverse direction and maintain speed. How many particle-particle collisions occur before the five particles occupy their original positions and are moving to the right?

In how many ways can a chess Queen move from one corner of the chess board to the opposite corner, moving closer to the goal square at every step?

What is the only positive integer $n$ such that there are exactly $n$ incongruent triangles with integer sides and perimeter $n$?

How many ways can you make one million dollars using any number of pennies, nickels, dimes, quarters, one-dollar bills, five-dollar bills, ten-dollar bills, twenty-dollar bills, fifty-dollar bills, and hundred-dollar bills?

What is noteworthy about the multisets $\{2,2\}$, $\{1,1,2,2,2,3,3,4\}$, and $\{1,1,1,2,2,3,3,3,4,6\}$? Can you characterize all multisets of positive integers with this property?

Monday, August 6, 2012

Four Spheres in a Tetrahedron

Four spheres of unit radius are contained in a regular tetrahedron in such a way that each is tangent to three faces of the tetrahedron and to the other three spheres. What is the side length of the tetrahedron?
This problem is from the book Beautiful Mathematics," published by the Mathematical Association of America.

Wednesday, August 1, 2012

A Square Root Calculation

Calculate $\sqrt{\phantom{XXXXXXXXXXXXX}} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! {(\underbrace{111\ldots1}_{\mbox{100 1's}})(1\underbrace{000\ldots0}_{\mbox{99 0's}}5)+1} \,\,.$

Solution: We can write the expression inside the square root as $\frac{1}{9}(10^{100}-1)(10^{100}+5)+1=\frac{(10^{100}+2)^2}{9}.$ Hence the answer is $(10^{100}+2)/3=\underbrace{3\ldots3}_{\mbox{$993$'s}}4$.

Sunday, July 22, 2012

Triangular Numbers in Base 9

A triangular number is a number of the form $n(n+1)/2$, where $n$ is a positive integer. Prove that $1$, $11$, $111$, $1111$, $\ldots$ are all triangular numbers in base $9$.

Solution: The proof is by mathematical induction. The first number, $1$, is triangular: $1=\frac{1(1+1)}{2}$. Each successive number is $9$ times the previous number (which puts a $0$ at the right) plus $1$. We claim that the operation of multiplying a number by $9$ and adding $1$ transforms any triangular number into another triangular number. Let's see: $9 \frac{n(n+1)}{2}+1=\frac{9n^2+9n+2}{2}=\frac{(3n+1)(3n+2)}{2}.$ This is a triangular number. And that completes the proof.

The operation multiply by $9$ and add $1$,'' used to produce new triangular numbers, can be generalized. If $k$ is a positive integer, multiplying any triangular number by $(2k+1)^2$ and adding $k(k+1)/2$ gives another triangular number. This is because of the identity $(2k+1)^2\frac{n(n+1)}{2}+\frac{k(k+1)}{2}=\frac{[(2k+1)n+k][(2k+1)n+k+1]}{2}.$ For example (with $k=2$), multiplying any triangular number by $25$ and adding $3$ gives another triangular number. It follows that $3$, $33$, $333$, $3333$, $\ldots$ are all triangular numbers in base $25$.