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Everything posted by bonanova

  1. Here are the placeholders for a long division, solvable, even with none of the digits filled in. The quotient has been placed to the side. It has a decimal point, not shown, and its last nine digits are repeating. Meaning, of course, the last row of X's replicates a previous row. Can you piece together the dividend? -------------- _________________ x x x x x x / x x x x x x x ( x x x x x x x x x x x x x x x x x x ----------- x x x x x x x x x x x x x ----------- x x x x x x x x x x x x x x ------------- x x x x x x x x x x x x x ------------- x x x x x x x x x x x x x x ------------- x x x x x x x x x x x x x x ------------- x x x x x x x x x x x x x x ------------- x x x x x x x x x x x x x x ------------- x x x x x x x x x x x x ----------- x x x x x x
  2. Dividend, please?

    Here are the placeholders for a long division, along with a single digit in the quotient. Can you piece together the dividend? x 7 x x x ---------------- x x x / x x x x x x x x x x x x ------- x x x x x x ----- x x x x x x x ----------- x x x x x x x x ------- - - - - -------
  3. Things are not going well at the Acme Company. Executive talent is hard to come by, and it is not cheap. Folks at the water cooler have no ideas, and the coffee-breakers can't imagine how to improve things either. But those who party around the teapot, they came up with something. They suggested to the Board that Acme promote the newest hire in the mail room and make him the CEO! We need to shake things up, but good. Qualifications, job experience, brains, judgment, integrity, these are all things of the past. Some were not so sure. Doesn't make sense at all, the old timers said. Almost like appointing some guy with orange complexion to be President. That's exactly the idea, said the tea-people. Turn things on end, let the bull loose in the china shop, and see what happens. Hey -- how could it be worse than what we have now? Not surprisingly, the debate was long and heated. Such a risk merited proof of possible gain, so the old guard posed a challenge: produce a concrete example of where the idea had been tried with incontrovertible benefit. In fact, make it mathematical. You know, something that might make a good BrainDen puzzle. We'll promote the mail room guy, they said, if you can show us an integer that doubles in value when its least significant digit is promoted to its most-significant position. That is, give us a number { some digits } q that has half the value of q { same digits }. That all happened last week, and now we're looking for the mail room guy. Was he promoted? Did the tea people find such a number? Is there one? We need a number or a proof that one does not exist. T.L.D.R. What number doubles in value by by moving its last digit to the first position (if there is one)?
  4. At a busy airport a conveyor belt stretches from the runway, where all the planes land, to the baggage claim area inside the terminal. At any given time it may contain hundreds of pieces of luggage, placed there at what we may consider to be random time intervals. Each bag has two neighbors, one of which is nearer to it than the other. Each segment of the belt is bounded by two bags, which may or may not be near neighbors (to each other.) On average, what fraction of the conveyor belt is not bounded by near-neighbors? Example: ----- belt segment bounded by near neighbors ===== belt segment not bounded by near neighbors ... --A-----B==========C---D--------E=============F---G-H---I-- ...
  5. Heads up

    Alternate approach to the ant-checkerboard problem asks: What is the probability of flipping 8 heads before 5 tails?
  6. Heads up

  7. Tempus fugit

    I was born at 11:35pm Central time in the United States on yesterday's date. The fact that I now live in the Eastern time zone might have moved my birthday to today's date. IDK. I don't know how these things work, nor in the grand scheme whether it even matters. I suspect it doesn't. The point of writing this, is the shock I had last night when friends took me out to dinner, good chicken and ribs btw, and one of them said to me, so you're __ years old now. My reaction was to say "no way" until I realized they had it right. And I thought once again about my mortality. My grandfather lived to within days of his 100th birthday, and I've always had it in the back of my mind that I would finish that journey in his honor -- even tho it might entail outliving certain government benefits. But one never knows. BrainDen community lost SuperPrismatic this year, at an age much less than mine at present. But not to be morbid. Que sera, sera, and let's celebrate the day, each day. I heard from all my children and grandchildren yesterday, and life is good.
  8. derivative brain teaser

    Looks like ...
  9. Next card is Red

    @CaptainEd shuffles a standard deck of playing cards and begins dealing the cards face up on the table. At any time @plainglazed can say Stop and bet $1 that the next card will be red. If he does not interrupt, the bet will automatically be placed on the last card. What is the best strategy? How much better than 50% can @plainglazed do?
  10. Heads up

    That was the first-cut answer in the ant-chessboard problem. The solution is easier to find if we (reasonably) assume that
  11. Balancing weights

  12. Breaking a three-way tie

    Al, Bert and Charlie are on the ballot for secretary of the local yacht club, and they finish in a three-way tie. To break it, they solicit the members' second choices, and again it's a three-way tie. Al notes that since the number of members is odd, a series of two-way votes would be decisive. As a reward for finding a way past the impasse, Al receives the bye and will take on the winner of a Bert vs Charlie vote to decide who gets the position as secretary. Assuming voter preferences do not change, what are the winning probabilities for the three gentlemen?
  13. I'm humbled. btw I sharpened (with humor) the flavor text and posted this on another favorite site. It will be interesting to see how quickly it's solved there.
  14. @Molly Mae gets the coveted bonanova gold star for finding an algorithm and getting the answer in less than one day. So now that the cat (algorithm) is out of the bag, so to speak, it should be trivial to find whether an integer can triple, (or quadruple, or quintuple) under the same manipulation. It's also interesting that his solution forms a ring that shows the doubling process also occurs for four other ending digits (not 1, so his is the smallest.) But which ones? Incidentally, this also shows that all such integers have 18 digits.
  15. a barrel of pickles

    Fun one.
  16. Three matches

    If we place four matches in the form of a square, they form 4 right angles. If we place them like a hash-tag (#) they form 16 right angles. If someone removes one match, can we still form 12 right angles? (No bending or breaking of the matches is allowed.)
  17. Baggage on a conveyor belt

    The answer probably does change with the nature of the distances.
  18. Ants on a checkerboard

    Two ants named Al and Bert sit at diagonal corners of a checkerboard and decide to change places. Al, at the lower left, walks randomly upward or to the right, and Bert, at the upper right, walks randomly downward or to the left. They follow the boundaries of the checkerboard squares. That is, except when following the extreme boundary of the checkerboard, their left and right feet always touch squares of opposite color. What is the probability of their meeting (1) if they walk at the same speed, or (2) if Al walks 3 times as fast as Bert?
  19. Ants on a checkerboard

    Regarding the 5 probabilities after 12 ant moves:
  20. Can you write down a 9-letter word that permits you to erase it, one letter at a time, such that after each erasure a valid (English) word remains? (As implied, the letter order remains the same throughout.) Clue: Example:
  21. making obtuse triangles

    Case by case results, inviting algorithm falsification, while limiting lengths to 10.
  22. Sorting out the bar bill

    A bunch of friends went to the sports bar and got a group rate on the drinks: $5/glass for wine, $2/glass for beer, and $1/glass for water. When we left, the waiter asked me to sort out the bill. There was enough uncertainty in what people remembered that I could not be precise. So we happily just threw in enough to cover the bill, which came to $293 and we went home. But it got me thinking. None of us had multiple glasses of the same beverage. The waiter said 106 glasses were used, once each. 18 of us did not drink water. 39 people had wine. I was certain that 9 of us were teetotalers. If I had known the sizes of just three classes of drinkers I could have figured out the bill, as it was, I could not. But it did occur to me that if those who drank beer and water but not wine were as many as possible, and if those who drank only wine were half as many as that, I could say the smallest number of us who drank all three. Can you?