This paper explores the trade-off between entanglement of a qubit with two other qubits. The authors show that the sum of the tangles between a qubit and each of two others cannot exceed the tangle between the qubit and the pair of the other two. This inequality is as strong as possible, as it can be satisfied by a quantum state. They define a "three-way tangle" that measures the entanglement of three qubits and is invariant under permutations of the qubits. Quantum entanglement is a key resource for quantum communication and information processing. The paper introduces the "tangle," a measure of entanglement related to the "entanglement of formation." The tangle is defined for a pair of qubits and is zero for unentangled states and one for completely entangled states. The paper shows that for a pure state of three qubits, the tangle between a qubit and each of the other two cannot exceed the tangle between the qubit and the pair of the other two. This result is extended to mixed states, where a minimum tangle is defined. The paper also defines a "residual tangle," which is the amount of entanglement between a qubit and the pair of the other two that cannot be accounted for by the entanglements with each of the other two. The residual tangle is invariant under permutations of the qubits and represents a collective property of the three qubits. The authors also define a "three-way tangle" that measures the entanglement of three qubits and is invariant under permutations. The paper concludes that the tangle is an optimal measure of entanglement with respect to the inequality derived. The results are extended to larger collections of qubits, and the authors conjecture that the corresponding inequality is valid for all pure states of n qubits. The work is supported by the National Science Foundation and the Isaac Newton Institute.This paper explores the trade-off between entanglement of a qubit with two other qubits. The authors show that the sum of the tangles between a qubit and each of two others cannot exceed the tangle between the qubit and the pair of the other two. This inequality is as strong as possible, as it can be satisfied by a quantum state. They define a "three-way tangle" that measures the entanglement of three qubits and is invariant under permutations of the qubits. Quantum entanglement is a key resource for quantum communication and information processing. The paper introduces the "tangle," a measure of entanglement related to the "entanglement of formation." The tangle is defined for a pair of qubits and is zero for unentangled states and one for completely entangled states. The paper shows that for a pure state of three qubits, the tangle between a qubit and each of the other two cannot exceed the tangle between the qubit and the pair of the other two. This result is extended to mixed states, where a minimum tangle is defined. The paper also defines a "residual tangle," which is the amount of entanglement between a qubit and the pair of the other two that cannot be accounted for by the entanglements with each of the other two. The residual tangle is invariant under permutations of the qubits and represents a collective property of the three qubits. The authors also define a "three-way tangle" that measures the entanglement of three qubits and is invariant under permutations. The paper concludes that the tangle is an optimal measure of entanglement with respect to the inequality derived. The results are extended to larger collections of qubits, and the authors conjecture that the corresponding inequality is valid for all pure states of n qubits. The work is supported by the National Science Foundation and the Isaac Newton Institute.