At what age must a Muslim fast and make up for the days they missed?

The pen of accountability (taklif) is only applied to the mature (baligh), for the Prophet, peace and blessings be upon him, said: "The pen has been lifted from three: from the sleeper until he awakens, from the insane until he recovers, and from the child until he reaches puberty." Puberty is attained by nocturnal emission, or the growth of pubic hair, or menstruation for females. If these signs do not appear, then the age of puberty is fifteen years according to the most authoritative opinion. It is incumbent upon the guardians of non-pubescent children to command them to fast so that they may become accustomed to it.