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The answer usually given to this question is "metal" and that answer is correct.

But, a metal is actually a material that conducts electricity well. The reason both conductivities, electrical and thermal, are high is that the "free electrons" that make a metal conduct electricity are also the mechanism for conducting heat.

Malleable means "easily shaped" and the question remains as to why metals are easily shaped.

Atoms in solids are held together by "bonds" formed by the electrons of the constituent atoms. (That is a long diversion to explain further.) The "bonds" in a metal are different than those in a generic materials because the electrons in those bonds are not strongly localized in the immediate vicinity of the individual atoms. These "metal" bonds are actually diffused throughout the metal. As a result, moving metallic atoms relative to one another does not require large distortions of the bonds and does not necessitate breaking bonds. The movement or material requires a general rearrangement of the electronic structure of the diffuse metallic bonds but that leaves the "bonds" in place.

Thus, gradual disruption of atomic arrangement associated with changing the shape does not result in the destruction of the microscopic arrangements of "bonds" as it does in a brittle material where cracks result.

A small caveat is necessary. Cracks do develop in metals, but the explanation here is one of degree and a qualitative description of the microscopic explanation of malleability.

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