10 pennies

Question:

Can you put the 10 pennies shown in Figure below into the three glasses in such a way that each glass contains an odd number of pennies?

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The first thing candidates need to understand is that, as stated, the problem is incapable of solution. Dividing 10 pennies among three glasses so each glass holds an odd number of pennies is impossible because any three odd numbers added together yields an odd number. The trick here is to change the arrangement of the glasses so a penny could count as being in more than one glass at a time. By placing an even number of pennies into a glass, an odd number of pennies into a second glass, and then placing the second glass into the first, both glasses contain an odd number of pennies (see Figure 4.2).

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Solution: A number of solutions are possible. One solution calls for 2 pennies in one glass, 3 pennies inside a glass within the first glass, and 5 pennies in the third glass.

Extra credit: How many times per day do the hour and minute hands of an analog clock form a right angle?

First, let’s consider the obvious answer. The hands of a clock form a right angle twice an hour (at a quarter to and a quarter past the hour). So with 24 hours in a day, that means the hands will form a right angle 48 times a day. Right? Not quite. Between 2:00 and 3:00 and then between 3:00 and 4:00, only one right angle is formed. Check it out yourself. So that means subtracting 4 (for both a.m. and p.m.) from the total.

Solution: 44 times.

 

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