Michael Reid’s Happy New Year Puzzle, 2017

Belatedly posted by permission of Michael Reid. Enjoy!

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

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.

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