Guessing the date

Mukta thinks of a date and keeps it secret. Shubham tells her: multiply the month by 5, add 6, multiply by 4, add 9, multiply by 5, then add the day. Mukta says 291, and Shubham replies, 'You thought of Republic Day, 26/01.'

Check her working with month 1 and day 26: 1 × 5 = 5; 5 + 6 = 11; 11 × 4 = 44; 44 + 9 = 53; 53 × 5 = 265; 265 + 26 = 291.

Now the algebra. Let the month be M and the day be D.

Look at what all those steps really did. The month has been multiplied by exactly 100, and a fixed 165 was added along the way. Nothing else.

So the trick is: subtract 165, and what is left is 100M + D. The day is at most 31, so it only needs two digits. The last two digits are the day, and whatever comes before them is the month.

From 291: 291 − 165 = 126. The last two digits are 26, so D = 26. What is left in front is 1, so M = 1. The 26th of January.

Mukta tries again and says 1390. Subtract 165: 1225. The last two digits are 25, the front is 12: the 25th of December.

You can make your own version. If you change the steps, the fixed number to subtract changes too. Keep the month multiplied by 100, and the month and day stay neatly apart.