When I was in high school until maybe a year ago, I have always been into math puzzles. I am not particularly good, but I thought that solving math puzzles were the ultimate goal, and I got myself thinking about them lately, while walking home around 2 AM after a day in the computer lab working on my thesis.
So I want to share a few of my favourites here:
In some countries, the shipping cost for a box of dimensions \( A \times B \times C \) will not be proportional to its volume, but rather to the sum $A + B + C$. (e.g. shipping a $1 \times 1 \times 27$ cm box costs 29CHF whereas a $3 \times 3 \times 3$ cm box cost only 9CHF.) Is there a situation where one can save money by packing one costly box inside another cheaper one?
Find the smallest positive integer that leaves a remainder of $1$ when divided by $2$, a remainder of $2$ when divided by $3$, a remainder of $3$ when divided by $4$, … , and a remainder of $9$ when divided by $10$
Suppose that we throw $2012$ (or $n$) fair dice at the same time. What is the probability that the sum of the values of the upturned faces of the dice is divisible by $5$?
Prove or disprove that if $n$ is odd and $0 \leq i \leq n − 1$, then
$$ \left( (-1)^i {{n-1}\choose{i}} \right) - 1 $$
is always a multiple of $n$.
I don’t remember the solutions anymore, but I’ll think about them on my next walk and post some of the solutions in the next post. I hope I can still solve them.