1. Yesterday was squareday – for everybody, but it meant more to me. Yesterday was 4/25/09 – all square numbers. In a given century this’ll happen 135 days out of 36,525 about 1 in 250 days – so not that big of a deal. But for me it marked my 36^th birthday – also a square number. This was the only day in my life when this will happen. In fact, for around 1 in 400 people this will only happen once. For everyone else it will never happen. Of course, if you were born on one of the 15 square-days in ’00 you should get at nine of these, maybe ten if you don’t smoke!
2. Oh, your birthday is all primes! Congrats. When will the special birthday come for you when your age and the year are both primes? Never. Sorry, if you were born in an odd-numbered year, you’ll turn an even age in odd years (primes must be odd), and an odd (potentially prime) age in even years.
3. What do you call the special class of numbers that individually are the difference between two consecutive square numbers? Let’s see. Let the first consecutive number be x and the second (x + 1). The difference in the squares is (x + 1)^2 – x^2 = y. Where y is one of the special class of numbers. Well, if you remember your FOIL method, y = 2x + 1. In order to ensure that both x and y are integers, the only stipulation is that y be an odd number. That’s it. All odd numbers can be created by the difference of two consecutive squares.
4. Trying to find a square-root without a calculator? Check this out this page. http://www.homeschoolmath.net/teaching/square-root-algorithm.php
5. So, some numbers have integer square roots (4, 25, 9, 36) but most have non-integer irrational number square roots. Is it possible to have a non-integer rational square root? I don’t think so. Coming back to the FOIL method, we could write an equation like: (a + x/y)* (a + x/y) = P. All of which are integers. Where a^2 < P (P is the number we’re taking the square root of); and x<y. A little manipulation gets us to:

   (P – a^2) = 2x/y + x^2/y^2
   with a little more manipulation we can get to 2xy + x^2 = Ky^2,

   Where K is some integer multiplier.
   Solving for x: x = y±√(4*y^2+4*K*y^2) = y ± 2*y*√(2*K)

   x = y(1 + 2√2K).

   Since √2 (square-root of 2) is irrational, no integer values of x,y or K will produce a rational number.

Long ago, when I started this blog, I wanted to release some new clock designs. I was motivated by the TokyoFlash approaches, and wanted to come up with some based on epicyclic movement. Why would one bother? Longer ago, I had difficulty reading an ‘analog’ clock. I was a child of the digital age, and preferred to the red blocky LEDs that simply spelled it out. I still find the hour and minute hand confusing sometime – squinting across the room – it’s easy to confuse the two hands.

So, in this first installment, I show the simplest design. It’s so simple that it may be deceiving how to read it. There is no hour hand. Instead the face of the clock moves to follow the minute hand. The minute hand is the most important anyway, right. Chances are when you look at a clock, you already know the hour, and you simply want to know the minute. In this clock (yes, I realize I need a better design – any offers?), the hour is indicated by the last number the minute hand passed. Get it? You can also download a Vista Gadget of this clock (remove the .zip from the end when you save), but it requires you to also install the following XAML extension: Windows Sidebar Styler.

Finally, I really would like a second hand to show some movement so that you know that the clock is working. So, in this modification, I’ve added an outer dial to show the seconds. The dial moves backwards and the exact second is indicated by the minute hand as well. In this way, your eyes need only go to where the minute hand is pointing to determine hour, minute, and second. Here’s the Vista Gadget for this one (again remove the .zip from the end when you save) (don’t forget the sidebar styler extension as well).

There are only a few magazines I read so thoroughly that I run out of articles before the replacement shows up on the newsstand, but Newsweek is one of them. The last page alternately written by Anna Quindlen and George Will is a favorite. While Ms. Quindlen is a superior writer (I’d kill for that clarity and flow in writing), I usually find George Will’s perspective refreshing and enlightnening.

However, this weeks foray into the interdependence of our economy as constrained by all other world economies misses the mark for me on one key issue. The anecdote of a wooden pencil being the product of four distinct parts that are based on raw material from four regions of the world is cute and awe-inspring, but it doesn’t happen the way Will and the economists he endorses say it does. He states, “goods … result from innumerable human actions but not from any human design”

Really?! All the artifacts around us just happen out of “emergent behavior”? You are missing a very important player in this – the designers – both the engineers and the industrial. Without them, the pencil would not exist. Perhaps this isn’t of interest to these pundits and economists, but we shouldn’t forget that inventor spirit. It’s what made this country great. Seriously, that sounds corny, but I’ll go further and say it wasn’t freedom of religion, or a “perfect” democracy, but the inventions that Americans bore. These inventions went on to change infrastructure, create jobs, and a quality of life beyond what is percievable by only examinig such economics. In the current crisis, we have shouldn’t forget this. Plus a career inventing things would be pretty damn fulfilling.

Last night I saw Return to Forever in concert. The opening date of their first tour in 25 years. Their nervousness and the crowd’s anticipation nearly stifled their astounding performance. But I must say, it was quite a thrill. Afterwards, I overheard fans fantasizing about seeing reunions of other jazz-fusion supergroups of that same era – Weather Report and Mahavishnu Orchestra. Thinking of that triumvirate lead me to this most dorky of all analogies. Return to Forever is to Microsoft as Mahavishnu is to Mac as Weather Report is to Linux. Let me explain.

Return to Forever is the least rebellious (some would say least cool). Their products are often overly complex and that sometimes complicates the central goal. Some product features are added to great effect others not so much (Hello Again?).

Mahavishnu Orchestra on the other hand consistently put out lethally brilliant pieces that appealed to hip rockers. It was, however, ruled by an overcontrolling tyrant.

Finally, Weather Report started out with creations that were free. It had a huge roster of brilliant but forgettable players that didn’t stick around for long. They had a slow evolution from eclectic pieces to more commercial stuff that, in the end, consistently missed opportunities to be truly innovative. And maybe Jaco is Google! Taking the group towards its cohesive peak and later breaking off to develop his own successful (and better) material.

Well that was fun, but I don’t suppose we’ll be hearing Celestial Terrestrial Commuters in an Itunes commercial anytime soon.

Dominid

• Required: 28 regular dominoes (0 to 6) and 2 people
• Objective: To have the lowest score after all dominoes have been played.

Like most domino games, the two players draw and add dominoes to a single structure of dominoes that takes form in the middle of the playing area. But as you can see from the figures, this shape can be built “up” as well as “out.” It is for this reason that standard dominoes that are exactly twice as long as they are wide are necessary in order to stack correctly. Also, the big difference in Dominid is that players place dominoes side-by-side as opposed to end-to-end. In placing a domino, the only constraint is that it borders two numbers (see Fig. 1 and Fig. 5) – it doesn’t have to be the same numbers. The issue is that one’s score is determined (in part) by the distance between the placed domino and the one it borders. By distance, I am referring to the total difference in the numbers. So, for example, the last domino placed by a player in Figure 1 is the top one (2 6). Since this is adjacent to (2 5), then user score goes up one point (1 pt. = 2 – 2 + 6 – 5). And yes, this is always positive – that’s why we call it a distance.	Figure 1: Unlike other domino games, the numbers don’t have to match, but your scorce will depend on how different your “placed” domino is from its neighbors.
Next up. Building upwards. The next player drew the doublet (5 5). She wants to place it next to the 5 & 4 to minimize how much her score goes up. She can actually place this right on top of the 5 & 4 as is shown in Figure 2. Here it gets a tad more complicated. Determining the distance to 5 & 4 is only one option, since it is also next to 2 & 1 and (6 4). So, she could choose to make her distance 7 or 2 respectively. But, of course, since she wants to minimize her score it is compared to the 5 & 4 below. In this way, more options are available when building up than building to the side. But there is one constraint: a domino cannot be placed directly over another. For example, this (5 5) domino could not be placed on top of the (6 4) in the first and second figures. You can only build up by straddling two other dominos that are at the same height.	Figure 2: Building on Figure 1, a 5-doublet is placed on top of the (5 4).
That’s the easy part. Now here comes the real strategic element. One player is assigned the odd numbers (1, 3, 5) and the other is assigned the even numbers (2, 4, 6). At the end of the game, the evens player can subtract off from their score any contiguous cluster of same even numbers, so long as that cluster is greater than 3. An example? Good idea. In Figure 3, all dominoes have been played. The evens-player current has 17 but since there is a cluster of three 4′s and three 6′s, the end score is 17 – 3 – 3 = 11. The odd player ends with a 15, and he also has two clusters of three: the row of 5′s and the 1′s to the right. His score is 9. Hence, “odd” wins since 9 is lower than 11. A cluster of three pieces is worth three points, a cluster of four is worth four, five is worth five, etc. A cluster of two (or one) is not worth any points. Okay, that the gist of it. Let me start over with a more detailed description.	Figure 3: A final configuration shows that “even” has 2 contiguous clusters of three numbers (4 and 6), and “odd” has two clusters (1 and 5).
Start: All pieces are placed face down on the table and shuffled (moved about randomly). Both players simultaneously draw a piece from the pile and place it face up (as in Figure 4). Add the total of each piece. The player with the higher value goes first. If the two tiles have the same total (e.g. (4 0) and (1 3)) than the player with the highest single value wins (4 0). The player with the losing domino must arrange the two side by side in the center of the table while the winning player draws three dominoes to start their hand. Then, the losing player draws their 3 dominoes. The winning player plays his first domino and then declares whether he want odds or evens. Note, this doesn’t change what you dominoes you play. Even though, you declare evens you will likely be forced to play many odds, since drawing new dominoes is a random affair. However, since your score in the end will depend on the clusters of the numbers you declared in the beginning, you will find yourself playing different strategies with these pieces. After each placement of a domino, draw a new one so that you always have 3 dominoes in your hand. It’s best to keep them a secret from the other player, otherwise they can use this knowledge against you. Players take turns until all dominoes are played. In the last few rounds, the draw pile will be empty so your hand will dwindle from 3 to 2 to 1 final piece. There will be countless places to play a piece at any given turn. The only limitation is that your pieces must border two other pieces and cannot completely cover another piece. Figure 5 illustrates these two forbidden cases. With each turn, players should announce their new score. You can think of the scoring like golf. You want to have the lowest score, so each move will cost you some “strokes” depending on how different your placed piece is from those around it. If your piece stacks on top of others, remember – you can take whatever pair of neighbors leads to the lowest score (as in the example of Figures 1 and 2). But it must be a neighboring pair – not one number to the left for the top digit and one to the right for the bottom digit. Okay, so that’s it for basic game play. In terms of strategy, that’s a whole other matter. You want to try to connect your numbers in clusters while you play so that at the end, you can take off a big chunk of points. You can also strive to cover your opponent’s numbers to break up his clusters. Since you can’t control what dominoes end up in your hand, you will often be playing odds even though you’re trying to build even clusters. You will also learn that doublets take on a particular advantage in this game. If possible, hold onto those until the end. This is the first attempt to write up these instructions, so feel free to post your questions.	Figure 4: First two pieces of a new game are simultaneously drawn by the two players.
	(a) (b) Figure 5: The only types of moves that are not allowed are (a) placing a piece completely over another, or (b) not being along side two other numbers.