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  1. Yesterday
  2. This is another puzzle where precise wording is important -- I'll try to get it right, but if anything is unclear, please ask ... I'll start out by saying that all the circles in this puzzle have the same radius, the aspect ratio of the rectangle is not specified and does not matter, and its size, relative to the size of the circles is only indirectly implied. Only the constraints stated in the puzzle should be assumed. I've drawn 17 circles that at least partially overlap a rectangle. Their centers all lie within the rectangle. None of the circles overlap or even touch any of the other circles. There is no room for an 18th circle to be added to the group. That is, the circles are drawn in such a way that even though there is space between them, it is impossible to draw another circle whose center lies within the rectangle that does not at least partially overlap one of the first 17 circles. That is all you know about the relative sizes of things. And it is enough information to answer the following question: First, let's erase the circles that I drew. Then I will paint the rectangle red and give you a large supply of opaque white circles. What is the smallest number of circles you will need to completely cover the rectangle? (so that no red will be showing.) The centers of the circles, again, must lie within the rectangle, but now, of course, the circles can overlap each other.
  3. Building cars

    So when OP says "build exactly one car a day (no more, no less)" it means you can build any number of cars in the interval [1 2) because "and to be clear a partial car is as good as not building a car." So if you built 1 1/3 cars on the first day it would count as "exactly 1 car," because it would pass 1/3 of a car to the second day, when you would then have to build any number of cars in the interval [2/3, 1 2/3)? In general, is it correct to believe that at the end of every nth day you must have built [n, n+1) cars, except for n=7, after which you must have exactly 7 cars?
  4. Born on a Wednesday

    This is absolutely something I should have been able to reason myself into. D'oh.
  5. Building cars

    The only day where you cannot have a partial car built is the 7th day. The other days must have a whole car built to meet your quota but you can have part of a car as long as you don't make two in a given day. You must build something each shift.
  6. Building cars

    I'm not understanding something about two shifts building exactly one car. A whole car (in one shift) or two half-cars (in two shifts) seem to be the only cases. Maybe spoiler one other possibility as a means of explaining? Thanks.
  7. Jelly beans join the clean plate club

    It must be symmetric about the NW-SE diagonal, so your figure show all the cases you computed. Nice, btw. Hint
  8. Building cars

    You are in charge of building cars, you are tasked to build exactly one car a day (no more, no less) and to be clear a partial car is as good as not building a car. Your shift is separated into two parts in which you could either build a whole, half, third, fourth, or fifth of a car in a given shift. By the end of the week you are to have built 7 cars with no partial cars left over. How many ways can this be done assuming a 7-day work week?
  9. Jelly beans join the clean plate club

    The program isn't a proof, it's just an application of a greedy algorithm that starts from states with an empty plate and works backwards to see whether every state could eventually reach one with an empty plate. It covered every possible state for up to 500 jelly beans, but it doesn't prove that a plate can always be cleared if you have something like 8x1012 jelly beans.
  10. Whodunit?

    @Thalia,
  11. Jelly beans join the clean plate club

    @plasmid Does the program imply a proof that it can always be done? Or is it a statement that no counterexample has yet been found? A proof could be a repeated procedure which after each application reduces the smallest number of beans on a plate. Does your algorithm always reduce the number on place C?
  12. Hats of three colors

    @ThunderCloud Nailed it. @Izzy Honorable mention
  13. Party time at Peter's and Paul's

    I think it boils down to that, but how to justify doing it?
  14. Born on a Wednesday

    @Izzy ... @Molly Mae ... you were halfway there!
  15. Hats of three colors

    I thinkā€¦
  16. Hats of three colors

  17. Party time at Peter's and Paul's

    @Izzy Yes.
  18. Hats of three colors

    @Izzy Very close. @Donald Cartmill OP restricts what each prisoner is allowed to say: Each prisoner must guess the color of his own hat, without having seen it, by saying one of the three colors
  19. Party time at Peter's and Paul's

    @Izzy As you say, the distribution is surprising. To be certain of this expected attendance at the smaller party, you might want to ...
  20. Jelly beans join the clean plate club

    I wrote a perl program to do what I said earlier...
  21. Hats of three colors

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