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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.

Direct answer

Chicken-rabbit cage formula

Use the assumption method: pretend every head is a chicken, then convert the extra legs into rabbits.

rabbits = (legs - 2 x heads) / 2; chickens = heads - rabbits
  1. Step 1

    Assume all chickens

    Start with 2 legs per head, so the baseline is heads x 2.

  2. Step 2

    Find extra legs

    Subtract the all-chickens baseline from the observed leg count.

  3. Step 3

    Divide by 2

    Each rabbit contributes 2 extra legs compared with a chicken, so extra legs / 2 gives rabbits.

Example: 10 heads, 28 legs

28 - 2 x 10 = 8 extra legs. 8 / 2 = 4 rabbits, and 10 - 4 = 6 chickens.

The interactive model below lets you keep the head count fixed and swap chickens into rabbits until the leg count matches.

Read the 4-method walkthrough

What this game shows Β· Chicken & Rabbit Cage (ιΈ‘ε…”εŒη¬Ό)

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 β€” 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 Γ— 2.
Extra legs
observed legs βˆ’ all-chickens legs. Each rabbit adds 2.
Rabbit count
extra legs Γ· 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

Olympiad thinking model

Who this demo helps, and where to practice next

Chicken & Rabbit Cage is built for students who need a visual way to decode multi-step puzzle structure. It gives the page a clear search purpose: learn the model, manipulate it, then continue into the matching grade-level practice.

Chicken & Rabbit Cage helps when a student can copy a procedure but cannot explain why it works. The demo slows the idea down into a visible model before sending the learner to guided missions.

Learning goals

  • If every head were a chicken, the cage would have heads x 2 legs.
  • Every rabbit has 2 extra legs compared with a chicken.
  • Extra legs divided by 2 gives the rabbit count; the remaining heads are chickens.

How to play

  1. 1 Fix one quantity and watch which quantity changes.
  2. 2 Write the hidden relationship in words before writing an equation.
  3. 3 Use the related grade topics to transfer the puzzle move into standard word problems.
FAQ

Chicken & rabbit cage, answered.

01 What is the chicken-rabbit cage problem (ιΈ‘ε…”εŒη¬Ό)? 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 Γ— 2 = 20 legs. Extra legs = 28 βˆ’ 20 = 8. Rabbits = 8 Γ· 2 = 4. Chickens = 10 βˆ’ 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 βˆ’ 2 = 2.

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