Dominoes are 2×1 rectangles. Assume you want to tile a rectangular room with dominoes and you don’t allow constellations where four dominoes meet – see the red x above. What is the smallest room that can not be tiled? We assume, of course, that the area of the room is an even number and that the width and the length are integers.
Problem source and illustration: Project Euler.
The more I want to get something done, the less I call it work. – Richard Bach
The two most common elements in the universe are Hydrogen and stupidity. – Harlan Ellison
Sept. 15 (Bloomberg) — Norwegian Labor Party Prime Minister Jens Stoltenbergbecame the first leader in 16 years to win re-election after steering the Nordic nation out of a recession and vowing to improve welfare.
Labor and its partners, the Socialist Left and the Center Party, secured 86 seats out of 169 seats in parliament, after 99.9 percent of the votes were counted. The opposition, fronted by the Progress Party and Conservative Party, won 83 mandates. – Read more
Innocent question: did the winning bloc get most votes? They must have, right, since they got the majority in the parliament and will form the new government?
They did not. The losing bloc got 40,000 more votes. At the election four years ago the losers got 20,000 more. From 1945 to 1961 Labour had the government, but never the majority of votes behind them.
I thought democracy was defined by ‘One man, one vote.’ If it is, few, if any, countries are democracies.
Why is that?
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Read more (in Norwegian): Here and here.
Television is more interesting than people. If it were not, we would have people standing in the corners of our rooms. – Alan Corenk
SOFIA (Reuters) – The draw of the same six winning numbers twice in a row in Bulgaria’s national lottery was a freak coincidence, officials said Thursday.
Sports Minister Svilen Neikov ordered an investigation after the numbers 4, 15, 23, 24, 35 and 42 were selected, in a different order, by a machine live on television on September 6 and 10. The results caused suspicions of manipulation.
An investigation found no wrongdoing in the draw or determining the winners, its chairman Konstantin Simeonov said.
“We cannot talk about any manipulation,” he said.
The chance of the same six numbers coming up twice in two consecutive rounds was one in more than 4 million but was not impossible, respected mathematician Michail Konstantinov has said. – Read more
We all know that 1 / (4 x 10^6) is different from zero, so Konstantinov is just stating the obvious. Or did he mean something more when he said that it was not impossible?
“This is happening for the first time in the 52-year history of the lottery. We are absolutely stunned to see such a freak coincidence but it did happen,” a spokeswoman said.
Here is today’s query. What is the chance that the same numbers would never be drawn in consecutive weeks during 52 years?
For almost every course one can find a small set of questions … questions that can be stated with the minimum of technical language, that are sufficiently striking to capture interest; that do not have trivial answers, and that manage to embody in their answers the important ideas of the subject.
P R Halmos 1975
Every math teachers have a set of beliefs. Here are some of mine. I believe that:
Last lesson I asked my students, 16 year olds doing a higher level IB math course, to find 1 + 2 + 3 + … + 10^6 for home work. Only one student found something. She took repeated differences of the sub problems, 1, 1+2, 1+2+3, … and found that 1 + 2 + 3 + … + n was a quadratic function. Using simultaneous equations she found it to be n^2/2 + n/2.
I was impressed and rewrote the result as n(n+1)/2. ‘When you have climbed a mountain you sometimes discover easier ways up. The way I have written the formula makes me think of triangles. For tomorrow, find another proof for the formula.’
No one found anything. That is, after waiting patiently for a few minutes someone volunteered a drawing of the triangular number 10. 4 dots at the base and 1 at the top. Like a pyramid. I asked if that gave them a clue.
It didn’t. So I added a similar triangle, but upside down, thereby creating a n x (n+1) parallellogram and from there everybody saw the proof.
Milking the situation further I said that part of the equation looked like an average, adding two numbers and dividing by two: (n + 1)/2. We looked at 1 + 2 + 3 + 4 and discovered it was the same as 2.5 + 2.5 + 2.5 + 2.5. Pairing 1 and 5, 2 and 3, we saw why. We had reached the result a third way similar to what supposedly Gauss did when he still wore knickers.
A fourth proof used mathematical induction using the domino analogy. We had also time to play the Towers of Hanoi in a flash version and discuss silly dialogues like: What is 3 x 5? I don’t know, but I think it is 5 more than 2 x 5, all the way down to 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1.
Several general principles were formulated and discussed. Solve a simpler problem if you can’t solve the original one. Look for patters. Guess. Prove. Look back. Are there other ways to solve the problem now that we have solved it once and know the answer. Sometimes it helps a lot if you already know what to prove. It may be easier to solve a more general problem.
Soon I will be inspected, or rather my teaching will be inspected. I have to provide a lesson plan one week in advance. Houston, I have a problem. I do not write lesson plans. I have a general idea about a problem to discuss, a definition, an algorithm, etc, but what will happen in my classroom I have no way of knowing beforehand. After the lesson I write ‘what we did’ notes, but I do not know how to hand those in a week in advance.
