# Mathematicians play hide and seek!

The ghosts of Eratosthenes and Euclid are playing hide and seek. They can hide anywhere in all of space and time, but the pair are in agreement to play somewhere on earth within the years 0 to 2000. Eratosthenes just found Euclid so now it's Eratosthenes's turn to hide. He scrawls out a note, hands it to Euclid and POOF! -- he spirits himself away to the hiding place. Dumbfounded, Euclid brings the note to you for help. The note reads:

29 a

129 ab

141 ab

826579 cbaa .

53 ca

I=29

In what day, and what place, is Eratosthenes probably hiding?

Hint 1:

The names are hints towards the same feature of this puzzle

Hint 2:

This riddle would be solvable without the letters, but it would be a bit harder

Hint 3:

Hint 4:

The second part is encrypted slightly differently to throw you off. The row following the colon represents two quantities, with which you can do what the puzzle tells you to do.

Message:

"I AM HIDING ATOP AN APPLE TREE. MULTIPLY: 1643"

How you get the message:

From Eratosthenes name we get the hint we are looking for primes. Take each number and decompose it into its unique prime factors so for example 129 becomes (3, 43). Based on the hint that I=29 we do the following. Find each prime numbers rank in the ordered set of primes with 2=0, 3=1, 5=2, 7=3, etc so (3,43) -> (1, 13). Then find the letter of the alphabet corresponding to that rank so (1, 13) -> (A, M). Finally we use the letters at the end of the line to let us know how to put them together. Since for 129 we have ab we know the lowest prime goes first followed by the next higher so the word is AM. Repeat this for all the rows.

How you get the date:

The hint tells us two "Multiply" so we are looking for at least two numbers, preferably primes. The first can be the 53 at the beginning of the line. Next using ca as a hint can get the digits that match the letters to give us 31, which just so happens to be a prime. Multiplying 53*31=1643 the year of Isaac Newton's birth.

Where he is:

He is visiting the Woolsthorp Apple Tree where Isaac Newton developed his theory of universal gravitation on January 4, 1643 the day of Isaac Newton's birth.

• Perfect answer, Barker. I thought of this encryption while falling asleep last night. Would you know if this encryption method is common, on SE? This was my first puzzle. I am impressed with the ingenuity of people here Commented Apr 13, 2018 at 4:00
• @CalebDevine I think the puzzle was pretty unique and clever. It pulled familiar elements (e.g. prime numbers, indicators for ordering, mapping numbers to letters) so it wasn't unapproachable, but it did it in a way I haven't seen before at least. It had reasonable clues too (e.g. Eratosthenes as a hint). Commented Apr 13, 2018 at 17:44
• My only criticism would be that the last part didn't tie into the rest of the puzzle and wasn't clued very well. It looked like it was similar to the rest of the puzzle, but it dropped the mapping between primes and letters and pretty much everything else the rest of the puzzle used. I ended up solving it basically by trail and error until I found a date that made sense for Newton. Core puzzle was fun though. Commented Apr 13, 2018 at 17:44
• Thanks for the detailed feedback! I should probably have put more time into the last part, not rushing to publish it at 1 or 2 am. Commented Apr 13, 2018 at 18:19