Belatedly posted by permission of Michael Reid. Enjoy!

Here are some interesting puzzles to start the New Year; hopefully they are not

tooeasy!1. Express 2017 as a quotient of palindromes.

2. (a) Are there two positive integers whose sum is 2017 and whose product

is a palindrome?

(b) Are there two positive integers whose difference is 2017 and whose

product is a palindrome?3. Is there a positive integer n such that both 2017 + n and

2017 n are palindromes?4. What is the smallest possible sum of the decimal digits of 2017 n ,

where n is …

(a) … a positive integer?

(b) … a prime number?

(c) … a palindrome?5. Consider the following two operations on a positive integer:

(i) replace a string of consecutive digits by its square,

(ii) if a string of consecutive digits is a perfect cube,

replace the string by its cube root.Neither the string being replaced, nor its replacement, may have

have “leading zeros”. For example, from 31416 , we may change it to

319616 , by squaring the 14 . From 71253 , we may change it to

753 by taking the cube root of 125 .(a) Starting from the number 2017 , what is the smallest number we

can reach with a sequence of these operations?

(b) What is the smallest number from which we can start, and reach

the number 2017 with a sequence of these operations?6. Find a list of positive rational numbers, q_1 , q_2 , … , q_n

whose product is 1 , and whose sum is 2017 . Make your list as

short as possible.

Extra Credit: Prove that you have the shortest possible list.