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?

2. Using only common mathematical symbols and the digits 2, 0, 1, and 6, make an expression that is exactly equal to 100.

3. Find a fraction with the following decimal equivalent.

4. How many positive integer factors does 2016 have?

5. What is the value of n if 1 + 2 + 3 + ··· + n = 2016?

6. Find 16 consecutive odd numbers that add up to 2016.

7. Create a 4 × 4 magic square in which the sum of each row, column, and diagonal is 2016.

8. Find a string of two or more consecutive integers for which the sum is 2016. How many such strings exist?

9. What is the value of the following series?

10. What is the units digit of 2²⁰¹⁶?

11. 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?

12. What is the value of the following expression, if x + 1/x = 2?

13. 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?

14. 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.)

15. 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?

16. What is the maximum possible product for a set of positive integers that have a sum of 2016?

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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:

• What is the binary representation of 2016?
• What is the base-4 representation of 2016?
• What is the base-8 representation of 2016?

Math Problems for 2016

“What homework do you have to do tonight?”

I ask my sons this question daily, when I’m trying to determine if they’ll need to spend the evening doing word study or completing a math worksheet, or if we’ll instead be able to waste our time watching The Muppets or, perhaps, pulling up the animated version of Bob and Doug Mackenzie’s 12 Days of Christmas on Dailymotion.

When I asked this question last night, though, the answer was surprising:

We have to do our reading, but we already completed your math problem.

My problem? I had no idea what this meant. So they explained:

It’s not a problem you gave us. It’s one we got from [our teacher], and it says, “This problem was written by Patrick Vennebush.”

I was puzzled, but then it dawned on me. I asked, “Does it have a monkey at the top with the word BrainTEASERS?”

“Yes!”

“Which problem?”

“It’s about the word CAT.”

I knew the problem immediately. It’s the Product Value 60 brainteaser from Illuminations:

Assign each letter a value equal to its position in the alphabet (A = 1, B = 2, C = 3, …). Then find the product value of a word by multiplying the values together. For example, CAT has a product value of 60, because C = 3, A = 1, T = 20, and 3 × 1 × 20 = 60.

How many other words can you find with a product value of 60?

As it turns out, there are 14 other words with a product value of 60. Don’t feel bad if you can’t find them all; while they’re all allowed in Scrabble™, the average person won’t recognize half of them.

You can see the full list and some definitions in this problem and solution PDF.

This problem resurfaced at the perfect time.

With 2016 just around the corner, no doubt many math teachers will present the following problem to students after winter break:

Find a mathematical expression for every whole number from 0 to 100, using only common mathematical symbols and the digits 2, 0, 1, and 6. (No other digits are allowed.)

And that’s not a bad problem. It gets even better if you require the digits to be used in order. For instance, you could make:

• 2 = 2⁰ + 1⁶
• 9 = 2 + 0 + 1 + 6
• 36 = (2 + 0 + 1)! × 6

But that problem is a bit played out. I’ve seen it used in classrooms every year since… well, since I used it in my classroom in 1995.

So here are two versions of a problem — the first one being for younger folks — using the year and based on the Product Value 60 problem above:

How many words can you find with a product value of 16?

How many words can you find with a product value of 2016?

There are 5 words that have a product value of 16 and 12 words that have a product value of 2016 (spoiler: those links will take you to images of the answers). As above, you may not recognize all of the words on those lists, but some will definitely be familiar.

7 Math Mistakes to be Aware Of

April is Math Awareness Month, and some things to be aware of this month — as well as the whole year through — are common math errors. Here are seven that show up frequently.

Incorrect Addition of Fractions. It’s common for kids to add fractions as follows:

And while that algorithm works for batting averages in baseball, it doesn’t work in most other places. More importantly, this mistake is often unaccompanied by reasoning. For example, a student who claims that 2/3 + 4/5 = 7/9 doesn’t realize that with each addend greater than 1/2, then the sum should be greater than 1. That lack of thought bothers me.

Cancellation of Digits, Not Factors. While it’s true that 16/64 = 1/4 and 19/95 = 1/5, students who think the algorithm involves cancelling digits may also argue that 13/39 = 1/9, and that just ain’t right.

Incorrect Distribution. This one takes a lot of forms. In middle school, kids will say that 4(2 + 3) = 8 + 3. As they get older, they apply the distributive property to exponents and claim that (3 + 4)² = 3² + 4² or, more generally, that (a + b)² = a² + b².

The Retail Law of Close Numbers. A large portion of the population will buy a shirt for $19.99 that they’d pass up if it had a price tag of $20.00. Even though the amounts only differ by one cent, a lesser digit in the tens place makes the price feel much lower. Crazy, but true.

Ignoring the Big Picture. If you are a driver who is interested primarily in speed (and less concerned with price, looks, fuel efficiency, or other factors), would you rather have a vehicle with 305 horsepower or one with 470 horsepower? If you chose the latter option, congratulations! While the owner of a sweet 305-hp Ford Mustang will be sitting at home and sipping a mint julep on his front porch, you’ll still be doing 30 mph on the highway in your Sherman tank.

Correlation Implies Causation. As ice cream sales increase, the number of drowning deaths increases, too. But that doesn’t mean that having an ice cream cone willl make you less likely to swim safely, even if you failed to heed your mother’s advice to wait 30 minutes after eating. It’s just that ice cream sales and swimming-related deaths increase in summer, both of which are to be expected.

Just because two things happen to coincide doesn’t mean that one is the direct (or even indirect) result of the other.

Percents Don’t Work That Way. A 20% decrease followed by a 20% increase does not return you to the initial value. If you invest $100 in a company, and it loses 20% the first year, your investment will then be worth $80. If it gains 20% the next year, you’ll now have $96. Uh-oh.

What common math error do you see frequently, and which one bothers you the most?

Two Simple Math Games

In his article “What Is the Name of This Game?” author John Mahoney discussed the mathematics of the following game:

Cards numbered 1-9 are placed face up on a table. Two players alternate picking up one card at a time. The winner is the first player who has exactly three cards with a sum of 15.

You can play this game with nine cards removed from a deck of cards, or you can play online by going to http://illuminations.nctm.org/deepseaduel. The online version is a one-player game, but it has modifications that use different numbers of cards, different values on the cards, and different required sums.

Can you find the winning strategy for this game? (Hint: The strategy is described in the linked article above.)

Here’s a modification of the game that seems interesting, too.

Use cards with the following numbers: 1, 2, 3, 4, 6, 9, 12, 18, 36. The winner is the first player who has exactly three cards with a product of 216.

The optimal strategy for this game is different than the strategy from the original game. Can you find it?

Note: For the original game, there are eight sets of three cards with a sum of 15:

{1, 5, 9}, {1, 6, 8}, {2, 4, 9}, {2, 5, 8},
{2, 6, 7}, {3, 4, 8}, {3, 5, 7}, {4, 5, 6}

For the modification, there are 36 sets of three cards with a product of 216.

Perhaps that fact will help you identify the optimal strategy.

What is Your Favorite Number?

The WordPress Post-A-Week Challenge sends me a daily topic idea to consider for blog posts. Often, the prompts are not appropriate for a math jokes blog. For instance, some recent prompts have been:

• Grab the nearest book (or website) to you right now. Jump to paragraph 3, second sentence. Write it in a post.
• How do you find your muse?
• If you could bring one fictional character to life for a day, who would you choose?

But today’s prompt landed in my wheelhouse:

What is your favorite number, and why?

When Art Benjamin appeared on the Colbert Report, he said that 2,520 was his favorite number when he was a kid. When Stephen Colbert asked him why, he replied, “It was the smallest number that was divisible by all the numbers from 1 through 10.”

Tonight, I asked my twin sons Alex and Eli what their favorite numbers are.

Eli: 5, 15, 55, because my favorite number is really 5, but 15 and 55 are triangular numbers that have 5’s in them.

Alex: 21, because my favorite numbers used to be 1 and 2, and because it’s the number of cards you deal when we play Uno (3 players, 7 cards each).

My favorite number is 153, for lots of reasons:

• It is the smallest non-trivial Armstrong (or narcissistic) number — that is, it is an n‑digit number that is equal to the sum of the nth powers of its digits: 1³ + 5³ + 3³ = 153.
• Its prime factors are 3 and 17, and my birthday is 3/17.
• It is a triangular number. (Consequently, it’s the sum of 1 + 2 + 3 + … + 17.) As 351 is also a triangular number, 153 is also a reversible triangular number.
• It is the sum of the first five factorials: 1! + 2! + 3! + 4! + 5! = 153.
• The sum of its digits is 9, and the sum of its proper divisors is 9².
• It is one of only six known truncated triangular numbers, which means that 1, 15 and 153 are all triangular numbers.

Mathematician John Baez claims that his favorite numbers are 5, 8, and 24.

Got a favorite number? Share it, as well as the reason it’s your favorite, in the comments.

Fun Factor

Two numbers were having a conversation about their social lives.

28: Did you hear that 284 broke up with 220?
6: I’m not surprised. He’s far from perfect. But at least their break-up was amicable. 
28: Yeah, well, I heard she started seeing 12.
6: Really? He doesn’t have abundant charm. Don’t you think 10 would be a better match for her?
28: I don’t know. He seems so solitary!

Speaking of factors, I learned a neat trick this weekend for finding the sum of the factors of a number. Before I share that, consider the method for determining how many factors a number has. Take the number 12, for instance. The prime factorization of 12 is:

12 = 2² × 3

The following array can be used to generate all of the factors of 12:

	20	21	22
30	1	2	4
31	3	6	12

It’s obvious from the array that there are six factors. But the trick is to notice that each factor in the array is made from a power of 2 times a power of 3 — that is, each factor is equal to 2^m × 3ⁿ, where 0 ≤ m ≤ 2 and 0 ≤ n ≤ 1. Since there are three possible values of m and two possible values for n, then there are 3 × 2 = 6 factors of 12.

In general, if the prime factorization of the number takes the form a^p × b^q × c^r, then the number of factors is (p + 1)(q + 1)(r + 1) for exponents p, q, and r. (The process could obviously be extended if there are more than three prime factors.)

But look at the array again. The sum of all factors of 12 is equal to sum of all products that occur within the array. However, there is an easy way to find that sum, by taking advantage of the distributive property. The sum of the powers of 2 along the top is 2⁰ + 2¹ + 2² = 7, and the sum of the powers of 3 along the left side is 3⁰ + 3¹ = 4. Consequently, the sum of all factors of 12 is equal to:

(2⁰ + 2¹ + 2²)(3⁰ + 3¹) = 7 × 4 = 28

In general, if the prime factorization of a number is a^p × b^q × c^r, then the sum of the factors is:

(a⁰ + a¹ + … + a^p)(b⁰ + b¹ + … + b^q)(c⁰ + c¹ + … + c^r)

And again, this could be extended if the number had more than three prime factors.

Cool, huh?

Confounding Factors

My colleague Julia is preparing a talk about factoring for an elementary audience, and she created the following problem to use as a warm-up:

Take a two‑digit number ab, and find the least common multiple of a, b, and ab. For example, if you take the number 35, then LCM(3, 5, 35) = 105. For which two‑digit number ab is LCM(a, b, ab) the greatest? (The notation ab is used to indicate the two‑digit number with tens digit a and units digit b, which is equal to 10a + b. This notation is used to distinguish the two‑digit number ab from the product ab.)

Here are some math jokes about factors:

What do you call an amount that exactly divides a recipe for a sweet confection?
A fudge factor.

What do algebra equations and British television have in common?
An X Factor.

Sadly, both of those are my original jokes. Sorry. To cleanse your palate, check out one of Randall Munroe’s original jokes about factoring: