The teammates notice a yellow weathervane pointing south on top of a sort of pyramid made of a triangle, a square, and a pentagon, each with an arrow pointing to one of its corners. Before they think to say why they came, they can’t help asking, “What’s that?” “Oh, it’s just my clock,” replies Coach Taylor. “Your clock? But how do you tell the time?” “It’s simple, really. At noon and midnight each day, the weathervane is red, and it and all three arrows point due north. Each hour the weathervane changes color from red to yellow to blue to red to yellow and so on, and it rotates 90 degrees clockwise. And every minute each of the polygons rotates so that it is back to the same position except with the arrow pointing to the next corner clockwise.” The students notice that right now the triangle’s arrow is pointing roughly southeast, the square’s arrow is pointing due west, and the pentagon’s arrow is pointing a bit north of east.
What time is it?
The students then get distracted by what’s written on the board: “(2x-1)² + 4(x+3)² = (2x+7)² where x>0 Eureka!” This makes them curious, and they ask, “Don’t you usually collect terms to get all of the ‘x’s on one side?” Coach Taylor replies, “Oh no, it’s much easier to solve this way.”
What is x?
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Solutions to week 81
Poor Pascal. If you follow Rascal’s rule for the first seven rows, you will get the results below, showing that this is not Pascal’s triangle.We can also immediately read off the middle entry of the third row (two) and the middle entry of the fifth row (five). But what about the 2,304th entry in the 4,609th row? Well, you can see the column 1,4,9 that begins with the first entry of the third row; that’s kind of suggestive, so let’s continue the triangle for four more rows using Rascal’s rule, to get:This appears to confirm our hunch: that column consists of the square numbers. Moreover, that column consists of the first entry of the third row, the second entry of the fifth row, the third entry of the seventh row, the fourth entry of the ninth row, the fifth entry of the eleventh row, and so on. So it will included the 2,304th entry in the 4,609th row, which will then be 2,304². Therefore, the product Rascal asks for is 2×5×2304² = 53,084,160.
Now how in the world did Rascal choose this number? The key lies in the 2,304. If you try to factor it, you see that it has a lot of factors of two. In fact, it is 28×3². That means the product Rascal asks for is 2×5×(28×3²)² = 217×34×51, encoding the date that Rascal asked the question in the exponents 17-4-1 — April Fool’s Day, 2017.
Rascal’s Grid. It’s best to figure out the second part of Rascal’s question first. If we write out some of the first eleven rows of the grid in this problem, we getfrom which we see that this is the same array of numbers as in the previous problem, simply rotated 45 degrees!
Since the triangle in the previous problem is obviously symmetric around the central vertical line just from the way it is defined, the grid in this problem must be symmetric around the main diagonal leading down and to the right from the top left corner. Therefore, the 57th entry in the 23rd row must be equal to the 57th entry in the 23rd row, so their difference is zero.
The real-life “rascal”(s) in this story are eighth graders Alif Anggoro, Eddy Liu, and Angus Tulloch, who recently published a mathematical research paper on this “rascal triangle” in the College Mathematics Journal. Congratulations to these young mathematical explorers!
Links to all of the puzzles and solutions are on the Complete Varsity Math page.
Come back next week for answers and more puzzles.