One roti cut into three unequal but familiar pieces — a half, a third and a sixth.
Why nothing else works: start from 1⁄3 + 1⁄3 + 1⁄3 = 1. To make the units different, one of them must be made bigger — and the only fractional unit bigger than 1⁄3 is 1⁄2. Now 1⁄2 + 1⁄4 + 1⁄4 = 1, and the only way to enlarge a 1⁄4 to something new is 1⁄3. The third piece is then forced: 1 − 1⁄2 − 1⁄3 = 1⁄6.
Q2.
Can you find four different fractional units that add up to 1?
Answer
Yes. This puzzle has exactly six solutions (the order of the four fractions does not matter).
#
Four different fractional units
Check with a common denominator
1
1/2 + 1/3 + 1/7 + 1/42
21/42 + 14/42 + 6/42 + 1/42 = 1
2
1/2 + 1/3 + 1/8 + 1/24
12/24 + 8/24 + 3/24 + 1/24 = 1
3
1/2 + 1/3 + 1/9 + 1/18
9/18 + 6/18 + 2/18 + 1/18 = 1
4
1/2 + 1/3 + 1/10 + 1/15
15/30 + 10/30 + 3/30 + 2/30 = 1
5
1/2 + 1/4 + 1/5 + 1/20
10/20 + 5/20 + 4/20 + 1/20 = 1
6
1/2 + 1/4 + 1/6 + 1/12
6/12 + 3/12 + 2/12 + 1/12 = 1
The easiest one to find is 1⁄2 + 1⁄4 + 1⁄6 + 1⁄12 = 1 — split the 1⁄3 of the three-piece answer into 1⁄4 + 1⁄12.
Did you know? Writing a number as a sum of different fractional units is called an Egyptian fraction. The chapter shows another one: 19⁄24 = 1⁄2 + 1⁄6 + 1⁄8. Check it: 12/24 + 4/24 + 3/24 = 19/24 ✔
How to hunt for them: the biggest unit must be 1⁄2 (four units each 1⁄3 or smaller cannot reach 1 once they are all different). Then find three different units adding to 1⁄2, and repeat the same reasoning.