Math Problems for 2023

Happy New Year!

I’m feeling lucky about 2023… but perhaps that’s because 7⁷ mod 7! = 2023.

But I’m also feeling lucky because there are many interesting problems that involve the number 2023. I’m a little late in getting this post out, but all of the following problems attempt to use the number 2023 in some interesting way. Thanks to Professor Harold Reiter from UNC‑Charlotte, who supplied the ideas for the first two problems; the rest are MJ4MF originals. Enjoy!

• For 2023, the sum of the digits of the year, times the square of the sum of the squares of the digits, is equal to the year itself. Yeah, that’s a mouthful; more concisely, (2 + 0 + 2 + 3) × (2² + 0² + 2² + 3²)² = 2023. Can you find the only other four-digit number abcd with the property that (a + b + c + d) × (a² + b² + c² + d²)² = abcd?
• If you place one dot inside an equilateral triangle and connect it to each vertex, you get three non-overlapping triangles. Similarly, if you place two dots in an equilateral triangle — and no subset contains three collinear dots — you can connect the dots to form five non-overlapping triangles. How many non-collinear dots must you place inside the triangle to get 2023 non-overlapping triangles?
• The sum of the digits of 2023 is 2 + 0 + 2 + 3 = 7, and 7 is a factor of 2023. For how many numbers in the 2020s is the sum of the digits a factor of the number?
• In how many ways can a 4 x 4 grid be covered with monominos and L-shaped triominos? One such covering is shown below.

• What is the sum of 1 + 3 + 5 + 7 + ··· + 2023?
• How long would it take you to count to 2023?
• Using only common mathematical symbols and operations and the digits 2, 0, 2, and 3, make an expression that is exactly equal to 100. (Bonus: make an expression using the four digits in order.)
• All possible four-digit numbers that can be made with the digits 2, 0, 2, and 3 are formed and arranged in ascending order. What is the first number in the list?
• Place addition or subtraction symbols between the cubes below to create a true equation:

9³     8³     7³     6³     5³     4³     3³     2³     1³ = 2023

• Find a fraction with the following decimal equivalent.

• How many positive integer factors does 2023 have?
• Create a 4 × 4 magic square in which the sum of each row, column, and diagonal is 2023.
• What is the units digit of 2023²⁰²³?
• What is the value of the following expression, if x + 1/x = 2?

• Each dimension of a rectangular box is an integer number of inches. The volume of the box is 2023 in³. What is the minimum possible surface area of the box?
• What is the maximum possible product for a set of positive integers that have a sum of 2023?