## 16 Math Problems for 2016

Yes, I know that I just posted some Math Problems for 2016 on December 19.

But I’ve decided to post some more for a variety of reasons:

• 2016 is cool.
• It’s a triangular number.
• It has lots of factors. (I’d tell you exactly how many, except that’s one of the problems below.)
• After writing the problems for that previous post, I just couldn’t control myself.
• It’s my blog, and I can do what I want.

People who write math problems for competitions (like me) love to be cheeky and include the year number in a problem, especially when any sufficiently large number will do. When the year number is critical to the success of a problem, well, that’s just a bonus. With that in mind, there are 16 problems below, each of which includes the number 2016.

A fully formatted version of these problems, complete with answer key, extensions, and solutions, is available for purchase through the link below:

16 Problems for 2016 — just \$1

Enjoy, and happy new year!

1. What is the sum of 2 + 4 + 6 + 8 + ··· + 2016?
1. Using only common mathematical symbols and the digits 2, 0, 1, and 6, make an expression that is exactly equal to 100.
1. Find a fraction with the following decimal equivalent.

$0.\overline{2016}$

1. How many positive integer factors does 2016 have?
1. What is the value of n if 1 + 2 + 3 + ··· + n = 2016?
1. Find 16 consecutive odd numbers that add up to 2016.
1. Create a 4 × 4 magic square in which the sum of each row, column, and diagonal is 2016.
1. Find a string of two or more consecutive integers for which the sum is 2016. How many such strings exist?
1. What is the value of the following series?

$\frac{1}{1} \times \frac{1}{2} + \frac{1}{2} \times \frac{1}{3} + \frac{1}{3} \times \frac{1}{4} + \frac{1}{4} \times \frac{1}{5} + \cdots + \frac{1}{2015} \times \frac{1}{2016}$

1. What is the units digit of 22016?
1. Some people attend a party, and everyone shakes everyone else’s hand. A total of 2016 handshakes occurred. How many people were at the party?
1. What is the value of the following expression, if x + 1/x = 2?

$x^\mathbf{2016} + \frac{1}{x^\mathbf{2016}} + \mathbf{2016}$

1. A number of distinct points were placed along the circumference of a circle. Each point was then connected to every other point, and a total of 2016 segments were formed. How many points were placed on the circle?
1. Let A = 1, B = 2, C = 3, …, Z = 26. Find a word for which the product of the letters is 2016. (This one may look familiar.)
1. Each dimension of a rectangular box is an integer number of inches. The volume of the box is 2016 in3. What is the least possible surface area of the box?
1. What is the maximum possible product for a set of positive integers that have a sum of 2016?

UPDATE: Bonus Material!

Special thanks to my friend Harold Reiter, who created the following 2016 problems for use as MathCounts practice:

• What is the smallest number N such that the product of the digits of N is 2016?
• What is the sum of the divisors of 2016?
• What is the product of the divisors of 2016? Express your answer as a product of prime numbers.
• Solve the following equation:

$\binom{n}{2} = 2016$

• What is the binary representation of 2016?
• What is the base-4 representation of 2016?
• What is the base-8 representation of 2016?
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## 3 Comments Add your own

• 1. Joshua  |  January 1, 2016 at 10:56 pm

Fun problems!

Problem 12 is a nice follow-up to this post by Mike Lawler.

• 2. PhxTechie  |  February 11, 2016 at 5:47 am

For #2, does 20 × (6-1) count?

• 3. venneblock  |  February 15, 2016 at 6:38 am

Sure does, and it’s simpler than the solution I had in mind. Well done!

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