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The playground joke was "thirty days hath September; all the rest I can't remember"

30 days hath Septober, April, June and no wonder; all the rest have peanut butter, except for Grandma who has a red wagon.

to me the point of the poem is that it reminds you that there are 3 classes (30,31,feb) and the rhythm tells how many months are in each class. Remembering 4 booleans for which 4 are the 30 day months is easier than remembering 12 arbitrary integers (unrealistic worst case uncompressed memorisation technique)

I've never heard of it and it seems completely useless unless you're good at rote memorization (in which case you probably wouldn't need it): not only do some months have the same number of syllables and rhyme with each other, but also the order of the months in the poem seems completely arbitrary.

The knuckle mnemonic is how I learned it and while it doesn't help with remembering February, it's at least fairly foolproof because it still works if you get directions mixed up.

Songs and poems are easier to remember, at least for most people.