# One Fair Coin and Three Choices

Posted by Jason Polak on 29. January 2013 · 2 comments · Categories: elementary, math · Tags: , ,

A few nights ago as I was drifting off to sleep I thought of the following puzzle: suppose you go out for ice cream and there are three flavours to choose from: passionfruit, coconut, and squid ink. You like all three equally, but can only choose one, and so you decide you want to make the choice randomly and with equal probability to each.

However, the only device you have to generate random numbers is a fair coin. So, how you do use your fair coin to choose between the three options of ice cream?

Of course, you can only use coin flips to make your choice. For instance, cutting the coin into three equal pieces, putting them in a bag to create a new stochastic process does not count.

# Nonnegative Sums of Rows and Columns

Posted by Jason Polak on 11. December 2011 · Write a comment · Categories: elementary · Tags: , ,

For any $n\times n$ matrix $A$ with real entries, is it possible to make the sum of each row and each column nonnegative just by multiplying rows and columns by $-1$? In other words, you are allowed to multiply any row or column by $-1$ and repeat a finite number of times.

My fellow office mate Kirill, who also has a math blog, gave me this problem a few weeks ago and I thought about it for a few minutes here and there. The solution is in the fourth paragraph, so if you’d like to think about it yourself stop here before you get close.
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