Heads and Legs: How Many Chickens, How Many Rabbits? - 9314 Toolbox - Guess first, let the answer surface

1. A lesson that asks before it answers

Most sites hand you the formula for the heads-and-legs puzzle. This page doesn't. It gives you the puzzle — 12 heads, 40 legs — and a slider that guesses the number of rabbits, starting at 0.

Every extra rabbit turns one chicken into a rabbit, and the leg count under the stage updates live. You walk to the answer yourself.

2. How it plays

Drag the slider and three chips update in real time: the current guess, the arithmetic, and a verdict. A wrong guess only says how far off you are and points at what to observe — it never hands you the number of swaps.

Stuck? Press the hint button: three levels, one at a time — look at the gap, then the per-swap change, then the method. Hints never contain the final answer. Every attempt is logged on the right; click any entry to jump back. Auto-enumerate walks through guesses one by one if you'd rather watch.

3. Think first, then compare

Before dragging, answer in your head: if all 12 were chickens, how many legs? How far from 40? What does one swap add? How many swaps close the gap?

Then check yourself: all chickens gives 24 legs — 16 short. Each rabbit adds 2, so 8 swaps: 8 rabbits and 4 chickens. The shortcut rabbits = (legs − 2 × heads) ÷ 2 is exactly this reasoning compressed into one step.

4. The moment you hit it

Nailing the answer triggers a small burst of confetti, and the banner retells your route: from 0 to 8 rabbits, legs 24 → 40, +2 each time. That straight line is the shared secret of enumeration and algebra.

You can also set 35 heads and 94 legs — the original puzzle from the ancient Chinese classic Sunzi Suanjing. Enumeration still gets there.

FAQ

Why must the leg count be even? Chickens have 2 legs, rabbits have 4; any mix is even. An odd total means no chicken-rabbit solution.

Is enumerating slow? The gap moves in steps of 2, so you can jump several swaps at once — that's the seed of the algebraic shortcut.

Is anything uploaded? No. Everything runs in your browser; comments stay in your own browser too.

Tool: Heads and Legs explorer. Also try the phone plan comparison lesson.

Chinese version: https://www.9314.cn/tools/ji-tu-tong-long/