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Ask the Wizard #354

What is the area of a regular pentagon with side length 1?

"Anonymous" .

Click the box below for the answer.

(sqrt(5)/4)*sqrt(5+2*sqrt(5)) = apx. 1.72047740058897.

Click the box below for a hint.

Pentagon hint

Consider the image above. AC = CD.

Here is my solution (PDF)

On a windless day, Charlie takes his canoe to the river and paddles up upstream. He always paddles at the same rate (in other words if we ignore the current, he would always go the same speed). A mile after launching, his hat falls in the river. Ten minutes after that, he realizes his hat is missing and immediately makes a u-turn to catch up to it downstream. Charlie catches up to his hat at the same place he launched.

How fast is the current?

"Anonymous" .

Click the button below for the answer.

3 miles per hour.

Here is my solution (PDF).

This question is asked and discussed in my forum at Wizard of Vegas.

The following question refers to a bar bet. A random U.S. serial number for any bill is used. The player selects any three digits from 0 to 9. He must select three different numbers. Wins are based on how many of these three selected digits appear at least once on the serial number.

For example, if the player selections 7-0-2 and the serial number is 22227755, then there are two matches. It does not matter that the 2 and 7 matched multiple times.

Here are the odds offered:

  • 3 matches: 5 to 1
  • 2 matches: 1 to 1
  • 1 match: 9 to 5
  • 0 matches: 15 to 1

What is the probability of winning and house edge of each bet?

"Anonymous" .

The following table shows the number of winning combinations, probability of winning, and house edge for each bet. The right column shows the house edge ranges from 7.4% to 8.5%.

Serial Number Bets

Matches Pays Combinations Probability House Edge
3 5 15,426,684 0.154267 0.074399
2 1 45,771,270 0.457713 0.084575
1 1.8 33,037,245 0.330372 0.074957
0 15 5,764,801 0.057648 0.077632