Cranes and Turtles Calculator
Solve the classic cranes-and-turtles word problem: find how many of each from the total heads and total legs, with a step-by-step solution and a check.
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Input
Enter the total number of heads and legs to find how many cranes and turtles there are.
Leg counts default to 2 for cranes and 4 for turtles. Change them freely to count any two kinds of things.
Result
Answer
Cranes 3animals
Turtles 5animals
Head count check
3 plus 5 makes 8
Leg count check
2 times 3 plus 4 times 5 makes 26
How to solve it
If every animal were a turtle, the legs would total 4 times 8, which is 32.
The difference from the actual 26 legs is 26 minus 32, which is -6.
Swapping one turtle for one crane changes the legs by 4 minus 2, which is -2.
Dividing the difference -6 by -2 gives 3 cranes.
Subtracting 3 cranes from the 8 heads leaves 5 turtles.
How it works
- Assume every animal is a turtle (the one with more legs) and compute the total legs for that case.
- Find the difference between the assumed total and the actual number of legs.
- Divide that difference by the gap in legs (turtle legs minus crane legs) to get the number of cranes.
- Subtract the number of cranes from the total heads to get the number of turtles.
- Leg counts default to 2 for cranes and 4 for turtles, but you can change them for any two-item problem.
- If the numbers do not give a whole-number answer, the tool reports that there is no valid solution.
Frequently asked questions
- How does this solve the cranes-and-turtles problem?
- It first assumes every animal is the one with more legs (the turtle) and finds that total number of legs, then takes the difference from the actual leg count. Dividing that difference by the gap in legs per animal (turtle legs minus crane legs) gives the number of cranes, and the rest of the heads are turtles.
- Can I use leg counts other than 2 and 4?
- Yes. Cranes default to 2 legs and turtles to 4, but you can freely change both values to model any two kinds of things, such as vehicles with different numbers of wheels.
- What happens if the numbers do not divide evenly?
- If the given heads and legs do not produce a whole-number solution, the tool reports that there is no valid answer. This also lets you check whether a head-and-leg combination is possible at all.
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