# Michael Reid’s Happy New Year Puzzles, 2018

Belatedly posted by permission of Michael Reid. Enjoy!

Here are some New Year’s puzzles to help start out 2018.

1. Arrange the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 in the
expression $a^b + c^d + e^f + g^h + i^j$ to make 2018.

2. (a) Express $2018 = p^q + r^s$ where p, q, r, s are primes.
(b) Express $2018 = p^q - r^s$ where p, q, r, s are primes.

3. (a) Is it possible to put the first 9 primes, 2, 3, 5, 7, 11, 13,
17, 19 and 23 into a 3×3 matrix that has determinant 2018?
(b) Is it possible to put the first 16 primes, 2, 3, 5, … , 53,
into a 4×4 matrix that has determinant 2018?

4. (a) Express 2018 = A / B using the fewest number of distinct
digits.
For example, the expression 7020622 / 3479 uses only seven
different digits. But it is possible to do better than this.
(b) Express $2018 = (A_1 \cdot A_2 \cdot ... \cdot A_m) / (B_1 \cdot B_2 \cdot ... \cdot B_n)$ using the fewest number of distinct digits.

# 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.

# Michael Reid’s Happy New Year Problems, 2020

Posted by permission of Michael Reid. Enjoy!

New Year’s Greetings!

Here are some fun puzzles to start the year.

1. Substitute the numbers 1, 2, … , 9 for the letters
a, b, … , i in the expression $a^b + c^d + (e + f + g - h)^i$
to get 2020.

2. Use the digits 1, 2, … , 9 in order, and any of the usual
arithmetic operations and parentheses to get a number that is
as close as possible to, but not exactly equal to 2020.

3. Express 2019/2020 as a sum of distinct Egyptian fractions,
i.e. $1 / n_1 + 1 / n_2 + ... + 1 / n_k$ for integers $0 < n_1 < n_2 < ... n_k < 202049$
(but 202049 is not square).

5. Make a 4×4 matrix of single-digit integers (0-9) with digits
2, 0, 2, 0 on the main diagonal, and having determinant 2020.
Is it possible to do it with a symmetric matrix?

If you liked this one, check out other puzzles ont this blog tagged with “Michael Reid”.