35 Heads, 94 Legs: A 1,500-Year-Old Puzzle, 3 Ways

Step by step

1. 8th grade: 2 equations. An old Chinese book: none.
   Eighth graders solve this with two equations. A Chinese book from about fifteen hundred years ago used none.
2. 35 heads, 94 legs. How many of each?
   A cage holds chickens and rabbits. There are thirty-five heads and ninety-four legs. How many of each?
3. From the Sunzi Suanjing (China) · Japan: cranes and turtles
   The puzzle comes from the Sunzi Suanjing, an old Chinese math book. Later, in Japan, it became cranes and turtles.
4. Way 1: all chickens → 35 × 2 = 70 legs
   Way one: pretend they're all chickens. Thirty-five chickens have seventy legs.
5. 94 − 70 = 24 legs short · a rabbit has 2 more
   But we need ninety-four legs. That's twenty-four legs short. And each rabbit has two more legs than a chicken.
6. 24 ÷ 2 = 12 rabbits · 35 − 12 = 23 chickens
   So swap chickens for rabbits, two legs at a time. Twenty-four divided by two is twelve rabbits. That leaves twenty-three chickens!
7. Way 2: everyone lifts half their legs
   Way two is the old book's trick. Every animal lifts half its legs. Chickens stand on one leg, and rabbits stand on two.
8. 94 ÷ 2 = 47 · 47 − 35 = 12 rabbits
   Now there are forty-seven legs on the ground. Each chicken counts once, and each rabbit counts twice. So forty-seven minus thirty-five is twelve rabbits!
9. Way 3: c + r = 35 · 2c + 4r = 94
   Way three is the school way. Let c be the chickens and r be the rabbits. c plus r is thirty-five, and two c plus four r is ninety-four.
10. 2r = 94 − 70 = 24 → r = 12
    Double the first line: two c plus two r is seventy. Take that away from the second line. Two r is twenty-four, so r is twelve!
11. All chickens first · every 2 extra legs = 1 rabbit
    Three ways, one idea. Start with all chickens, and every two extra legs means one rabbit.
12. On a test: c + r = 35 · 2c + 4r = 94 → r = 12, c = 23
    On a test, write it like this. Name the unknowns. Write one equation for heads, and one for legs. Double the first one, then subtract. Solve, and answer in a full sentence.