## The Puzzle Archives, II

This week, I’ll continue with some more problems from the contests for the 2014 conference of the International Group for Mathematical Creativity and Giftedness.  We’ll look at problems from the Intermediate Contest today.  Recall that the first three problems on all contests were the same; you can find them here.

The first problem I’ll share is a “ball and urn” problem.  These are a staple of mathematical contests everywhere.

You have 20 identical red balls and 14 identical green balls. You wish to put them into two baskets — one brown basket, and one yellow basket. In how many different ways can you do this if the number of green balls in either basket is less than the number of red balls?

Another popular puzzle idea is to write a problem or two which involve the year of the contest — in this case, 2014.

A positive integer is said to be fortunate if it is either divisible by 14, or contains the two adjacent digits “14” (in that order). How many fortunate integers n are there between 1 and 2014, inclusive?

The other two problems from the contest I’ll share with you today are from other contests shared with me by my colleagues.

In the figure below, the perimeters of three rectangles are given. You also know that the shaded rectangle is in fact a square. What is the perimeter of the rectangle in the lower left-hand corner? I very much like this last problem.  It’s one of those problems that when you first look at it, it seems totally impossible — how could you consider all multiples of 23?  Nonetheless, there is a way to look at it and find the correct solution.  Can you find it?

Multiples of 23 have various digit sums. For example, 46 has digit sum 10, while 8 x 23 = 184 has digit sum 13. What is the smallest possible digit sum among all multiples of 23?

You can read more to see the solutions to these puzzles.  Enjoy!

## The Puzzle Archives, I

In going through some folders in my office the other day, I came across some sets of mathematics puzzles I wrote for a conference of the International Group for Mathematical Creativity and Giftedness in 2014.  Teachers of mathematics of all levels attended, from elementary school to university.  The organizing committee (which included me) thought it might be fun to have some mathematical activity that conference attendees could participate in.

So I and my colleagues created three levels of contests — Beginning, Intermediate, and Advanced — since it seemed that it would be difficult to create a single contest that everyone could enjoy.  But I did include three problems that were the same at every level, so all participants could talk about some aspect of the contests with each other.

Participants had a few days to get as many answers as they could, and we even had books for prizes!  Many remarked how much they enjoyed working out these puzzles.

Now this conference took place before I started writing my blog.   I have written several similar contests over the years for various audiences, and so I thought it would be nice to share some of my favorite puzzles from the contests with you.  And so The Puzzle Archives are born!

First, I’ll share the three puzzles common to all three contests.  I needed to create some puzzles which were fun, and didn’t require any specialized mathematical knowledge.  As I’m a fan of cryptarithms and the conference took place in Denver, I created the following puzzle.  Here, no letter stands for the digit “0.” For the next puzzle, all you need to do is complete the magic square using the even numbers from 2 to 32.  Each row, column, and diagonal should add up to the same number.  There are two solutions to this puzzle — and so you need to find them both! And of course, I had to include one of my favorite types of puzzles, a CrossNumber puzzle.  Remember, no entry in a CrossNumber puzzle can begin with “0.” I also included a few geometry problems, staples of any math contest.  For the first one, you need to find the area of the smallest circle you could fit the following figure into.  Both triangles are equilateral; the smaller has side length 1 and the larger has side length 2. And for the second one, you need to find the radius of the larger circle.  You are given that the smaller circle has a diameter of 2 units, and the sides of the square are 2 units long.  Moreover, the smaller circle is tangent to the square at the midpoint of its top edge, and is also tangent to the larger circle. The last two problems I’ll share from this contest are number puzzles.  The first is a word problem, which I’ll include verbatim from the contest itself.

Tom and Jerry each have a bag of marbles. Tom says, “Hey, Jerry. I have four different colors of marbles in my bag. And the number of each is a different perfect square!” Jerry says, “Wow, Tom! I have four different colors of marbles, too, but the number of each of mine is a different perfect cube!”

If Tom and Jerry have the same total number of marbles, what is the least number of marbles they can have?

And finally, another cryptarithm, but with a twist.  In the following multiplication problem, F, I, N, and D represent different digits, and the x‘s can represent any digit.  Your job is to find the number F I N D. (And yes, you have enough information to solve the puzzle!) Happy solving!  You can read more to see the solutions; I didn’t want to just put them at the bottom in case you accidentally saw any answers.  I hope you enjoy this new thread!

(Note:  The FIND puzzle was from a collection of problems shared by a colleague.  The first geometry problem may have come from elsewhere, but after four years, I can’t quite remember….)
Continue reading The Puzzle Archives, I