One day in math class, poor Chloe watched her teacher assemble a cube from 27 playing blocks. At home that afternoon she attempted to make the same structure from her modest set of blocks. She found that she was three blocks short. One entire row on the top level was vacant.
Chloe’s classmate Bubby MacFarlane has an enormous bag of blocks. In the middle of his living room, he set about the task of building the world’s largest cube ever made by a six-year-old. Instead of attaining success, he had the biggest tantrum ever when he too ended up with a vacant row on the top level. Poor Chloe comforted Bubby the next day in school.
We don’t know how many blocks Bubby owned, but like Chloe’s 24 it is a multiple of six. Can you show this with a picture or a little algebra?
Abby, Bill, and Carol were playing catch. They stood at the imaginary corners of an imaginary triangle so that they all would have the same distance to throw the ball. Math is all in your head, and these kids were using theirs.
Darryl joined the group. They noticed that they could not find a formation on level ground which gave them all the same distance to throw. (I don’t believe there is one.) The four settled on a square formation where there were only two distinct distances involved, one along the sides of the square, and the other between opposite corners (the diagonal distance). Incredible as it may seem, Carol, at home that evening, drew up five other formations that the four friends could have assumed, and still maintained only two different throwing distances. Can you sketch them?
BONUS QUESTION: Fast-forward to the future and we’re looking at four kids in personal space flying suits, playing laser tag and keeping a singular distance apart. What is the nature of their formation?
I have a game for you to try with an opponent. On a square array of dots, you will take turns choosing and circling one. The loser is the player who chooses the fourth corner of a perfect square, and is caught by the opponent. Start with a small grid of 25 dots, say, and see how long the game takes. Then gradually switch to larger, more challenging arrays. Caution: squares may be tilted and thus hard to spot, as in the examples in Figure 1. Your challenge: pretend that Figure 2 shows a game in progress, and that its your turn. I think there remains just one safe play for you. Can you find it, and tell me what row and column it is in?
solved by Sean McL.
The Lackwicks, Jim and Karen, are very excited about their $600 tax rebate check which is due in the mail any day. They plan to spend the full amount at heretoday.com, their favorite online shopping website. Heretoday is offering a Rebate Blowout Sale, and the only restriction is a one hundred item limit for the shopping cart. Jim and Karen were going to purchase three things in quantity (at least one of each): Carrot-Patch Kids for $18 each; Bubble Bunnies for $12 each; and Gummy Hares for $3 apiece. Describe all the ways in which the Lackwicks can purchase a total of 100 of these items for exactly six hundred dollars.
Solved by Lou C., Dane B., Matt E.
Most people have heard the legend of Pygmalion, the hermit sculptor, who made a statue of a woman so beautiful that he fell in love with it. In furtherance of the Cause of Love, Aphrodite made the statue come to life in the arms of the artist. Many have not heard the rest of the story.
Galatea, the woman, became enamored as much with sculpting as with Pygmalion. She made a request of the goddess: that after studying sculpture for a year, she would make statues of one man and one woman, and Aphrodite would bring them to life (and Love). Aphrodite approved the deal, and added that each new living creature, after studying for a year, would be allowed to create new, albeit somewhat ceramic, life; the females could sculpt one of each sex, and the males could create one female statue (after which they would lose interest anyway). No person could create another statue after that, but crocks and ashtrays were OK. This benefit was only fair since none could procreate in the normal fashion, each having a very undifferentiated system of internal organs. Aphrodite further declared that, if her rules were adhered to, all the artists would be immortal.
If you're keeping score of creations each year, that's one for Pygmalion, two for Galatea in the second year, three by her creations (male, female, female) in the third year, and so on. How many lives were created in the 25th year, what was the cumulative total of immortals, and what do you think happened to this colony?
[SOLVED by Lou C.]
I begin with a three-digit number. I multiply it by itself, and I observe that the product ends with the three digits I started with in order. I find that any power of the original will end in those same three digits. Can you tell me the number? There are two solutions, and neither one is 000 or 001.
solved by: Sean McL., Matt E.
A blind cowboy drove twelve head o' cattle into town (don't ask). At the corral a ranch hand herded them into four vacant pens. The cowboy asked, "Are you sure you got all twelve?" "Don't worry, there's twelve," replied the hand, ungrammatically. "How many are in each pen?" the cowboy asked. Gettin' a mite irritated, the hand counted heads in the four pens and snidely told the cowboy the product of those four numbers. Scratchin' his head, the cowboy asked "Is there a single cow in any pen?" "NO" said the hand, shortly. After that the cowboy knew exactly how many were in each of the pens. Do you know? Can you figur' it? Who'll be first to spill the beans?
[SOLVED by Vilasini B.]
My grandpa once told me "The day before yesterday I was 52 years old, and next year I'll turn 55. He never lied. Can you guess his birthday and the day he spoke to me?
Solution: Grandpa's birthday is on Dec 31. He is speaking on January 1st, the day after he turned 53. So the day before yesterday he was "only" 52. At the end of the year in which he made this statement, he will turn 54. Next year he will turn 55. -Ian Cowan
Let's start with something that may be more familiar: a magic square. A magic square is a square grid filled with numbers that possess the magic property, that is, the numbers in each row, column, and diagonal add to the magic sum. Perhaps an example will be de-mystifying.
In this square of order (size) 3, the magic sum is 51. You may have noticed that consecutive numbers were used. In such cases, when the order and the lowest number are known, it is possible to calculate the magic sum. Can the reader predict this sum for an order-five square of consecutive whole numbers beginning with 8?
In addition, you may have observed that the middle number of the set is located in the center of the square. This is an important factor in getting the diagonal totals in agreement with those of the rows and columns. What is the middle number of the 5x5 square mentioned above?
I offer two challenges this time: can you create the order-5 magic square which contains numbers from eight through thirty-two? I read, but I don't quite believe, that there are over fourteen million different solutions, so yours will probably be unique.
[SOLVED by Jim B., Vilasini B., Gerald H.]
Or would you rather do a cube? Place the numbers 1 through 27 in the three square grids below, in an arrangement which makes the cube, formed by stacking the squares, a magic one. You need to have equals sums in all the rows, columns, ranks (such as DEF), and the four main diagonals (such as JKL). Again there are multiple solutions.
[Two solutions submitted by Lou C.]
Gus and Freda were on one of their Saturday adventures, this time walking along a railroad track. They came to a tunnel and decided to walk through, even though they knew they shouldn't. After they had walked 2/5 (two-fifths) of the way through the tunnel, they heard a train whistle blow in the distance. In their fright the children began running in opposite directions. They both ran at a speed of 15 miles an hour. Each child barely got out of the tunnel in the nick of time. How fast was the train traveling?
solved by: Sean McL.
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