BrainDen.com - Brain Teasers # bonanova

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## Everything posted by bonanova

1. ## Forming a regular hexagon from a lattice

2. In the all-digital future, X and O are banished from the game of tic-tac-toe. They are replaced by 1 and 0, the the result of such a game might look like this: 1 | 0 | 1 - + - + - 0 | 1 | 1 - + - + - 0 | 1 | 0 Under the usual rules that require getting 3-in-a-row, it would be a draw. But this is the digital age, and there are different rules for winning. If we sum the eight rows of three numbers we get 2, 2, 1 (horizontally) 1, 2, 2 (vertically) and 2, 2 (diagonally). Six of the sums are even, and two are odd. The final parity of the board is thus even, and the game is said to have an even outcome. If there were more odd sums than even, the game would have an odd outcome. If there were four even (and therefore four odd) sums, the game would have a neutral outcome. The game is played as follows: The winner of a fair-coin toss (call him player A) chooses whether to play first or second. The other player (call her player B) decides whether she wants an odd, even, or neutral game outcome. On each turn, a player places his choice of either a 1 or a 0 on any unoccupied place on the grid. As in normal tic-tac-toe, players alternate turns; but here on each turn a player may play either a 0 or a 1. When the places are filled, the board is examined to determine whether it is odd, even or neutral. If the final board parity matches player B's choice, player B wins; otherwise player A wins. The questions to answer are: Is there an advantage to winning the coin toss? Is there a winning strategy for either player?
3. A while ago, about a clock with indistinguishable hour and minute hands and asked at what times of day, between the hours of noon and midnight, it was impossible to unambiguously determine the time. The hands moved continuously. This puzzle asks a related question. At what times of day, between the hours of noon and midnight, is it impossible to distinguish the hands of such a clock from those of its mirror image? Clearly noon is one of these times, but not in general thereafter -- since the clock's hands will move clockwise while the hands of its mirror image will move counterclockwise.
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• • 4. By connecting the eight vertices of an octagon (with horizontal, vertical and 45-degree lines, ignoring the perimeter) we define eight rows having either four or five points of intersection. Place sixteen consecutive (positive) integers on these points such that The sums along all eight rows are the same. The common sum is the smallest possible. A harder problem is to create the largest possible common sum.
5. Not quite. See Rainman's post for a strong clue.
6. Yeah... ... I was trying to revisit an But I couldn't find it (till just now) and did a bad job of describing the problem.
7. 1 3 1 2 2 8 3 3 1 4 3 0 5 3 1 6 3 0 7 3 1 8 3 . . . .?
8. Make it easier: Solve for t.
9. With a nod to try this: Let there be an infinite sequence of exponentiation t = xx... I don't know whether that could be expressed with up arrows or not, but Rainman's arrows reminded me of this. Anyway, there is an infinite sequence of higher and higher order exponents in this expression. Show that if t = 2 then also t = 4.
10. So optimal means minimize the sum: floors+doors+ keys.
11. What is the relation of the ten floors to optimality? Is optimal fewest operations or shortest time? Does it take time to travel between floors? Are there n elevators ? Are you initially inside your basement office? If so can you open its door without first identifying its key and using it to get out? This is a nice puzzle.
12. Sounds like a homework problem. Percentages are nice - they are invariant to scale. So here's a clue: what is the percentage change in the length of the ruler?
13. ## Three Planet Galaxy and Stock Market Chaos

14. ## Three Planet Galaxy and Stock Market Chaos

15. Aaaargh. Up arrows are famously non intuitive to think about. I'm thinking nonetheless.