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Do Unto Caesar

At the beginning of the night three poker players; Alan, Bernie and Craig had money in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of them won $1 200. What were the assets of the players at the beginning of the evening?

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Oh for the Mathematics of Yesteryear

A garrison of 600 men has just enough bread ... but, with the news that the enemy was planning an attack... How many ounces of bread a day must each man in the garrison be allowed, to hold out 45 days against the siege of the enemy?

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Hello Again

Anne completes a circuit around a circular track in 40 seconds. Brenda runs in the opposite direction and meets Anne every 15 seconds. How long does it take Brenda to run around the track?

Square Pegs

Age 11 to 14 Challenge Level:

For the square peg: if the diameter of the hole is $2r$ then the side of the square is $r\sqrt{2}$. The fraction of the area of the cross section of the hole taken up by the peg is $(r\sqrt{2})^2/\pi{r^2} = 2/\pi$.
For the round peg: if the diameter of the peg is $2r$ then the side of the square is $2r$. The fraction of the area of the cross section of the hole taken up by the peg is $\frac{\pi{r^2}}{4r^2} = \frac{\pi}{4}$.

We know that $\pi^2 > 8$ so it follows that the round peg is a better fit as it takes up more of the hole because $\frac{\pi}{4} > \frac{2}{\pi}$.