遊び, 遊ぶ, and 遊戯: Naming Play and Describing Play in Progress
Compare a stone game, absorbed play, park and block play, and festival or group games. Read activity nouns, the verb, and past play in progress without fixed age or formality rules.
An invented game and absorbed play
遊び (あそび) means play or amusement. 「子どもたちは石を使った遊びを考えた。」 — “The children invented a game using stones.” The children invented a stone game, without specifying its rules or winners.
「彼は遊びに夢中で約束の時間を忘れた。」 — “He became absorbed in play and forgot the time of his appointment.” What he forgot while absorbed in play was the appointment's time. The sentence does not report how late he was or a resulting conflict.
Park play and ongoing block play
遊ぶ (あそぶ) means to play. 「放課後に友達と公園で遊ぶ。」 — “I play in the park with my friends after school.” The example describes playing in the park with friends after school in a nonpast statement.
「子どもたちは積み木で楽しそうに遊んでいた。」 — “The children were happily playing with blocks.” 遊んでいた describes the children's play in progress at that past time. Read their happy block play without changing it into a report that play had finished.
A traditional game and a group game
遊戯 (ゆうぎ) means a game or recreation. 「祭りでは昔から伝わる遊戯が披露された。」 — “A traditional game was demonstrated at the festival.” A traditional game was demonstrated at a festival. This example does not establish an exclusive ceremonial or formal use for the word.
「子どもたちは広場で集団遊戯を楽しんだ。」 — “The children enjoyed a group game in the square.” The children enjoyed a group game in the square. 遊び and 遊戯 name activities, while 遊ぶ describes playing; these examples do not restrict any of the words to one age group.
Does the children's 遊んでいた with blocks mean that they had finished playing?
Show the answer and why
No. It describes happy play in progress at that past time. The festival and group-game examples of 遊戯 likewise do not establish a restriction to adults or formal occasions.

