Possible on the first day.
Your pet will have a thought bubble pop up when he is "bored" with the icon for one of the Play games in it. Simply hold your hand over the bubble and you will play the game.
Do this 10 times and it will pop.
I did this in between "Chase time" where you run/walk in a circle and your pet chases you. I was going for the "Chase Me!" award for chasing your pet 10 minutes. There was typically a bubble within moments of finishing a chase sequence.