A blog for enthusiastic math-lovers!

Real Analysis Proofs

Here are two great items to prove about our number systems (they may seem intuitive at first, but rigorously proving them can be challenging). See if you can answer them.

1) Prove that between any two real numbers there exists a rational number. Similarly, prove that between any two real numbers there exists an irrational number

Hint: Use Archimedean property.

2) Prove that the union of a finite number of countable sets is countable.


Hey guys,

Check out the new Android app that I wrote, “Codester: A Brainteaser Puzzle”. It’s lots of fun and it teaches programming concepts!

Find it here: https://play.google.com/store/apps/details?id=com.codesterapp&hl=en  or just type in “Codester” in the Play Store search.

It works on any Android device, tablet, or smartphone. Let me know what you think, thanks.

Clever Geometry Problem

Another interesting problem!

Suppose you have a quadrilateral ABCD where AB is parallel to CD and such that the length of DA = AB = BC and BD = DC.
What are the angles of this quadrilateral?

Hint: Obviously, a trigonometry solution is possible with some work. But there’s another way, without writing down a single number! Keep in mind your shapes and rules about certain, specific edges being equal to one another as in this problem.

Triangle Problem

My math teacher showed me this interesting problem. It may seem simple at first but see if you can find a generalization to it.

How many triangles with integer side lengths have perimeter 10?

Alright, this problem isn’t too difficult. But what if we wanted to see how many triangles with integer side lengths have perimeter x for any x? Is there a nice way to generalize this such that the answer is of the form f(x) for some function f? Is there a way to prove that this generalization is true for any integer x?

Last week, Wild about Math interviewed me for the “Inspired by Math Podcast series”.

Check it out: http://wildaboutmath.com/2013/03/31/gili-rusak-mary-okeeffe-inspired-by-math-27/

Next Puzzle Hunt!

My next math puzzle hunt will be on March 24! Hope everyone can make it!

In the mean time… another problem I came across:

How many positive numbers between 1 and 100000000 have the same number of 0’s, 1’s, 2’s, and 3’s as digits such that no leading zeros are allowed?

Interestingly enough this was a computer science problem, but see if you find a solution using strictly math.

Halloween Puzzle Hunt

It is yet another wonderful fall Sunday afternoon in the Capital District. It is partly sunny and cool. The chill is in the air and the cold wind blows as innuendo to the cold days yet to come. The dry foliage leaves are scatters on the road and the sidewalk, twirling in the wind. It is a perfect afternoon to our second Puzzle Hunt program.

Today we are honored to have a special guest with us. Dr. Stephen Wolfram, the CEO of Wolfram Matematica, is here today. I met Dr. Wolfram at the Math Prize for Girls competition this year. While talking to him there, I explained the idea of our Puzzle Hunt group and he graciously agreed to sponsor our event. With the kind help of Ms. Carol Cronin from the Wolfram Alpha office we secured the prizes in the Puzzle Hunt Chest all courtesy of Wolfram Mathematica.

My co-organizer today was Catherine Wolfram, Dr. Wolfram’s daughter. I met her at the Math Prize for Girls competition. She expressed her interest in the Puzzle Hunt events and wanted to organize an event in her local town in Concord. She wanted to gain experience and see how this program worked. When I ask her to help me in this Puzzle Hunt event she eagerly agreed. Dr. Wolfram and Catherine drove all the way from Concord to Schenectady, NY to be with us today. I was so exciting about this opportunity.

When Catherine arrived the similarity between us was amazing: Short brown Uggs boots, skinny jeans, orange t-shirt (so we decided since out theme is Halloween see below) long, brown hair. Two enthusiastic teenagers, experienced in math, ready to instill the love of mathematics in our diverse group of younger girls. It is really amazing.

Dr. Mary O’Keeffe our Math Circle advisor, whose vision was coming true with this Puzzle Hunt Series was also here excited as ever. Mrs. Alexandra Shmidt was also there, and brought of course a wonderful tasty ginger bread cake, ready to help as usual.

In the ice breaker activity which is building geometric figures like tetrahedrons and I saw Dr. Wolfram engaged with the girls. Enthusiastically, he explained to them the concepts and showed them what you can do with the Wolfram Mathematica Program on his laptop. This was an added advantage to this event. The girls were really engaged.

For this Puzzle Hunt meeting I chose a different format than the first one. The first one consisted of a set of problems that the girls needed to solve into a crossword puzzle. The girls worked as a whole to solve those problems.

This time I wanted to convey new mathematics concepts to the girls. I wanted their knowledge to grow. Still in the spirit of the Ken Fan’s Puzzle Hunt Idea that the groups share its common knowledge toward one common goals of solving math problems, I decided to split the girls to four study groups. Of course, these were not a competitive groups, and I emphasized this from the start. I shuffled the girls according to their age and knowledge. I wanted them to form friendships and mutual work with each other, even if they aren’t so familiar with each other.

Since it is the weekend after Halloween, Catherine and I decided the we would continue the fun and work on three Halloween themes: Trick or Treat explaining the idea of summations and triangular numbers, working with MM Candies – explaining the prime numbers, and Pumpkin Geometry – exploring the Geometry of Spheres and 3-dimensional solid figures. Each theme had a problem set of its own.

Each group got the same problem set. The format of this Puzzle Hunt was like this: each group solved each problem and wrote it on the board. Each answer needed to be either checked or challenged by the other groups. After each successful solution of a problem set, the girls would get one magic number to their combination lock. After they will have all three magic number, they will open the puzzle hunt box.

The first theme of the summations was the most challenging to the girls and I was glad that I put it first. First we really imitated a trick or treat action; one girl from each group had to pick candies from the other groups as follows: 1 from the first table, 2 from the second table, 3 from the third table, 4 from the fourth table. The second girl picked 1 from the first table, 3 from the second table, 5 from the third table, 7 from the fourth table. The third girl picked 2 from the first table, 4 from the second table, 6 from the third table, and from the fourth table. Lastly the fourth girl picked 1 from the first table, 2 from the second, 4 from the third, and 8 from the fourth.

I explained to them the concept of the triangular numbers and gave them a formula page with different formulas of different series and the problem set. It was really challenging and not so easy. You could see that the girls were at first, a little intimated by the concept, wearing a “what is this about?” expression on their faces. But then with the help of Catherine and I and consulting each other, they really trying hard to tackled those problems. The answers started to flow to the board. Some are unanimous, to some we have two answers and we need to check them again. The dynamic is great. You could see enthusiasm to solve those problems and the ambition to succeed in this. Questions 7 and 8 are the hardest. In the end of this part I explain them on the board. Good Job! The girls had the first magic number to their lock!

The Second concept was Catherine’s idea. It was explaining Prime numbers. She asked the girls to build rectangles from their MM packages according to the MM color. She explained how the rectangles that could be made using the MM’s represented composite numbers while the ones that couldn’t represented the prime numbers. She also explained Euclid’s proof for why there are infinite prime numbers.

Another problem set. After about forty minutes, the girls successfully solved all the problems and received the second number for the combination lock!

The third activity was geometry of sphere; The girls got to hold a real pumpkin (which we considered a perfect sphere). They really measured it with rulers which helped them really feel and understand the concepts of radius, diameter and volume. Again a problem set to work on; this time with a real challenge because they needed to prove one question. The dynamic on the board was great. Again some challenges, some agreements and two very nice proves appeared on the board.

So now that we have all three magic numbers we have to open the chest.

This is a task for itself. It looks like you need to know this skill also, not just solving hard math problems! But after years of practice with the locker at the junior and high school you become an expert in this too, no doubt! After two fruitless attempts to open the chest it was finally opened! The prizes were great and the girls had a lot of fun. They will surely have some challenging stuff to do later at home with those prizes.

We were finished by now and it was time for the refreshments: fruits, fall cookies, ginger bread cake, bread sticks and of course…….. candies!

This Puzzle Hunt was a great learning experience and a great opportunity for expending the knowledge of the girls and learning new concept.

Great job everyone!!!

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