Another quick puzzle from Math E Magic (by Raymond Blum, Adam Hart-Davis, Bob Longe and Derrick Niederman).
"Find three distinct integers, a, b and c, such that 1/a + 1/b + 1/c = 1".
Firstly: does one of 1/a, 1/b or 1/c have to be 1/2?
Yes. Without 1/2, the largest number we can obtain is 1/3 + 1/4 + 1/5 = 47/60 (0.78333). So the first number has to be 2.
Now, what do we obtain by trying 1/2 + 1/3 + 1/4 (the next simplest solution)?
Answer: 13/12. So we're going to need something a little smaller. 1/2 + 1/4 plus anything else is going to be too small, so we'll progress with 1/2 + 1/3.
1/2 + 1/3 = 5/6, which immediately highlights the simple solution, that 1/2 + 1/3 + 1/6 = 1, since 1/2 = 3/6 and 1/3 = 2/6.
This was in the trickier section of the Mathemagic book, and it appealed to me as it can be solved by logic and reason, instead of pure trial and error (which, in my view, is what "hard" sometimes means in maths puzzle books :-) )
"Find three distinct integers, a, b and c, such that 1/a + 1/b + 1/c = 1".
Firstly: does one of 1/a, 1/b or 1/c have to be 1/2?
Yes. Without 1/2, the largest number we can obtain is 1/3 + 1/4 + 1/5 = 47/60 (0.78333). So the first number has to be 2.
Now, what do we obtain by trying 1/2 + 1/3 + 1/4 (the next simplest solution)?
Answer: 13/12. So we're going to need something a little smaller. 1/2 + 1/4 plus anything else is going to be too small, so we'll progress with 1/2 + 1/3.
1/2 + 1/3 = 5/6, which immediately highlights the simple solution, that 1/2 + 1/3 + 1/6 = 1, since 1/2 = 3/6 and 1/3 = 2/6.
This was in the trickier section of the Mathemagic book, and it appealed to me as it can be solved by logic and reason, instead of pure trial and error (which, in my view, is what "hard" sometimes means in maths puzzle books :-) )
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