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.
Other answers (if the integers don't have to be distinct) is 1/2 + 1/4 + 1/4 = 1, or the almost trivial 1/3 + 1/3 + 1/3 = 1. Yes, I know. Boring :-)
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 :-) )
Solve 1/a + 1/b + 1/c = 1 for unique a, b, c
Solving Magic Triangles
and the slightly more complex Magic Hexagons
"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.
Other answers (if the integers don't have to be distinct) is 1/2 + 1/4 + 1/4 = 1, or the almost trivial 1/3 + 1/3 + 1/3 = 1. Yes, I know. Boring :-)
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 :-) )
A few other articles in the Mathemagic Series:
Arrange nine coins into ten straight linesSolve 1/a + 1/b + 1/c = 1 for unique a, b, c
Solving Magic Triangles
and the slightly more complex Magic Hexagons
If you've found this puzzle interesting, can I recommend some of my other posts which have a similar theme?
Snakes and Ladders (the Collatz Conjecture)
Crafty Calculator Calculations (finding numerical anagrams with five digits)
More Multiplications (finding numerical anagrams, four digits)
Over and Out (reduce large numbers to zero as quickly as possible)
Calculator Games: Front to Back
Calculator Games: Up, up and away with Ulam sequences
Calculator Games: The Kaprekar Constant
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