<s>[INST] 10 pairs of identical socks. Each pair of socks has a distinct color. 10 of your friends give you one sock from their favorite pair. No friend gives you a sock that matches the color of any sock they have given you before. No sock given to you matches the color of any sock that has previously been given to you. Your friends want you to return all 10 socks in pairs. Since each friend only has socks with one color on them, the only way this can be done is if there is a sock color that occurs exactly 5 times (so two pairs to each of five friends) in the 10 socks that you got. If we could color-code each sock, then you know that the only way this can be done is if each of the 10 friends gave you a different colored sock. This means that the only way this can be done is to have all 10 colors in the set of 10 socks that you got from friends. So each of the 10 friends gave you a sock of a different color. And there are no other colors other than those 10 in all the socks. Hence there is only 1 sock (1 friend) colored as each of the ten colors. So each of the 10 friends gave you one sock colored as each of the ten colors. Since each friend only has one color on their socks, then once you give sock color a to friend a, each of the remaining 9 friends gave you sock color b or sock color c etc. It doesn't matter which one as long as you give these socks out so that each subsequent friend gets one of his own sock colors to complete his pair. So you can hand back all 10 socks.[/INST]