Inquiry AI
🐔

Chicken & Rabbit Cage

Heads fixed, legs reveal the mix

A classic Olympiad model game: keep the head count fixed, swap one chicken into one rabbit, and watch the leg count jump by two. The all-chickens assumption makes the hidden mix visible.

What this game shows · Chicken & Rabbit Cage (鸡兔同笼)

The chicken-rabbit cage puzzle hides a mix you cannot see: only the head count and the leg count are known. This game fixes the heads and lets you swap a chicken into a rabbit one at a time — every swap adds exactly two legs, so the hidden mix becomes a counting game you can run by hand.

All-chickens baseline
if every head were a chicken, legs would equal heads × 2.
Extra legs
observed legs − all-chickens legs. Each rabbit adds 2.
Rabbit count
extra legs ÷ 2. The remaining heads are chickens.

Aligned with CCSS 4.OA.A.3 (multi-step word problems). Recommended for Grades 4–6 as an Olympiad warm-up.

Olympiad model lab

Chicken & Rabbit Cage

Heads stay fixed. Every swap changes the leg count by two.

Heads10
Legs28
Chickens6
Rabbits4
Cage mix10 heads locked
Each chicken-to-rabbit swap keeps the same head count and adds exactly 2 legs.
+2+2+2+2
FAQ

Chicken & rabbit cage, answered.

01 What is the chicken-rabbit cage problem (鸡兔同笼)? Olympiad

A 1,500-year-old Chinese Olympiad puzzle: a cage holds chickens and rabbits. You see the total heads and total legs but not the mix. Find how many of each.

02 What is the assumption method? Assumption

Pretend every animal is a chicken (the lower-leg case). Observed legs minus that baseline = extra legs from rabbits. Divide by 2 (the leg gap per swap) to get the rabbit count.

03 Worked example: 10 heads, 28 legs? 4 rabbits

All-chickens baseline = 10 × 2 = 20 legs. Extra legs = 28 − 20 = 8. Rabbits = 8 ÷ 2 = 4. Chickens = 10 − 4 = 6.

04 Why does each swap add exactly two legs? +2 per swap

A chicken has 2 legs and a rabbit has 4. Swapping one for the other keeps the heads constant but changes the legs by 4 − 2 = 2.

Related Fun Math games

Explore more Fun Math →