A problem from February's contest

There was an interesting problem on the last contest that required factoring to solve. In what I present here I have modified the problem slightly from the original, but if can solve this, you can solve the other.

We were given a rectangle divided into 4 smaller rectangles. The smaller rectangles all had integer sides, but they were not of equal size.

Three of the inner rectangles had areas given. The problem was to figure out the perimeter of the whole figure.

(The graphic is not necessarily to scale, so don't rely on that to solve it!)

One trick to beware of is that you may become focused on finding the area of the fourth rectangle. Although the process is essentially the same, don't forget at the end of your efforts to answer the question that was asked. In this case, we are looking for perimeter.

If you want to check your answer here, the perimeter you get should be 46.

Contest review plus sequencing games

Contest review

Today we went over the contest from November. It was tough! As usual, most people who took the contest nationally got 0, 1, or 2 of the 5 questions.

I won't list the problems here, because they are proprietary, but I'll categorise them.

The first was one of those rearrangement / calculation problems we talked about the first day. You can blast through it as it is, or you can rearrange it to make your calculations a little faster.

The second problem was a sequence puzzle where some clues are given and then you deduce the order of some objects. More on this below!

The third problem was to figure out a three digit number based on some clues about the digits, including divisibility.

The fourth involved calculating the area of a shape surrounding some other shapes.

The fifth was a cryptarithm. You can find plenty of examples of those at http://www.cryptarithms.com

We'll do more of all of those throughout the year.

Sequencing logic puzzles

This kind of puzzle is really fun. There are lots of these on the LSAT — the standardised test for getting into law school. To practise these, I got some related puzzled from the Mindware game LogicLinks. They have samples from the four leveled books:

• Level A
• Level B
• Level C
• Level D

We worked through these until we ran out of time!

Divisibility by 3 and a Contest

Sums and differences again: remainders

Last time we talked about using sums and differences to tell if a number is divisible by 7.

For example, 781 is not divisible by 7: $781=770+11$, and we know that 770 is divisible by 7 and that 11 is not.

A similar kind of logic works with remainders.

• What is the remainder (R) when you divide 465 by 7?

  $465=420+45$.

  $45÷7$ → R3.

• What is the remainder if you divide (13 + 283) by 7?

• Method 1: $13+283=296=280+16$. $16÷7$ → R2
• Method 2: $13÷7$ → R6; $283÷7$ → R3; $6+3=9$; $9÷7$ → R2, so $(13+283)÷7$ → R2

The divisibility by 3 "trick":

If the digits in a number sum to a multiple of 3, then the number is divisible by 3.

How do we know this works?

First, see if there are any obvious exceptions: does it work for the first few multiples of 3 you can think of? How about the first few non-multiples of 3?

Okay, so you may have persuaded yourself that there is something to it, but does this prove it always works?

Of course not!

Here's the reason it works:

$10÷3$ → R1

So if you have, for example 54, $54=50+4$. You know that 50 will have a remainder of 1 when divided by 3, and that will happen 5 times. As far as remainders are concerned, this is just like adding 5.

We had to stop here to start...

Our first contest

which was really fun!

We'll go over the solutions, not next week, but the week after.

First Day!

Today at Math Olympiad we talked about 3 things:

• What does it mean to be good at math?
• What is Math Olympiad?
• Factors / Divisibility / Primes

Here is a quiz about being good at math:

Q. Is being good at math something you just naturally are or aren't?

A. Some people find math easier than other people, but the way you get good at math is by practising.

Q. If I make a lot or mistakes, or don't "get it" right away, does that mean I'm bad at math?

A. No! Learning a new math concept or technique takes practice.  Even professional mathematicians make mistakes sometimes and take time to develop their skills.

Q. I am a girl.  Is it weird for me to like math?  Am I naturally not as good at math because I'm a girl?

A. Absolutely not.  It's normal for girls to enjoy and excel at math.

If you believe you are good or bad at something, that can be a self-fulfilling prophecy, because if you think you are good at it, you will like it and keep trying, but if you think you are bad at it, you will stop trying.  Don't fall for the trap of believing you are bad at math!


What is Math Olympiad?

Math Olympiad is a problem solving program.  We meet once a week and practise solving fun math problems.  Once a month we try our hands at contests.

The contests always consist of 5 problems.  They have a single answer, and regardless of how well or poorly you approached the problem, your score on that problem is just 0 or 1, depending on whether you got it wrong or right.

Q. If I get only 1 or 2, or even no questions right, does that mean I'm bad at math?

A. Getting the right answer is fun and satisfying, but getting it wrong is okay, too.  The problems are designed to be tough!  Instead of comparing yourself to others, try to figure out how you can improve.

A Sample Problem

We looked an old contest, and tried the first problem.  The first problem often looks like a lot of arithmetic, but turns out to be easy if you rearrange it.

For example, we looked at a problem that asked:

What is the value of:

55 - 44 +11 - 22 + 33 - 33 + 22 - 11 + 44 - 55.

See if you can spot the trick that turns this into a very easy problem!
If you don't see it, don't worry.  There are usually many ways to solve a problem, and the best one to use is the one that makes most sense to you.

Factoring and Primes

A lot of Math Olympiad problems are based on knowing how to find factors.

A factor is just a fancy word for a natural number that divides evenly into another.
(A natural number is a counting number: 1, 2, 3, 4, ....; not a negative, decimal, or fraction)

So, for example, the factors of 6 are 1, 2, 3, and 6.

A natural number that has exactly 2 factors (itself and 1) is called prime.

For example, 2, 3, and 17 are prime.  15 is not prime, because it has 1, 3, 5 and 15 as factors.

1 is not prime.  It has only 1 factor.

How can you tell if a number is prime?  We checked 51 to see if it was prime by seeing if any number was a factor, starting with 2.

Checking all the numbers to see if they are factors can take a lot of time, especially if the number is big, or if we have a lot of numbers to check.

We made a Sieve of Eratosthenes to generate primes below 100.  Wikipedia has a nice animation of exactly what we did by hand.

Next time


Next time we will do more with factors, and talk about how to recognise when a number is divisible by other numbers.

Please join us!