Uncle Bob's Puzzle Corner
Sundry Puzzles Solutions
    1. The Bubby Cube – the “Bubby” shape can be reconfigured so that the number of unit cubes is the product of three consecutive numbers which always contains the factors 2 and 3.
    2. A Game of Catch – Hint: make use of the shapes of equilateral and isosceles triangles, a rhombus and the regular pentagon. In space, the children can form a tetrahedron to keep a single distance apart.
    3. Avoid-a-Square – Hint: The dot in Row 1, Col 2 is not the solution. It completes a square formed by moving one dot clockwise from each outer corner. You will have to screen other possibilities carefully for the one correct solution.
    4. The Consumer's Dilemma – one solution is 8, 20, and 72 Carrot Patch Kids, bunnies and hares, respectively. See if you can find all other solutions.
    5. Lost Art Colony – the yearly sculpture totals follow the Fibonacci sequaence beginning: 1, 2, 3, 5, 8, 13, 21, 34, 55, .... The partial sums of this sequence track the running totals of creations, and they are 1, 3, 6, 11, 19, 32, 53, and so on. Notice that the running total for a given year is two less than the number of new sculptures produced two years later. Can you prove that this is so? Does it check for your year 25 and year 27 figures?
    6. Narcissistic Powers – 376 and 625 will have their higher powers end in those selfsame digits.
    7. The Blind Cowboy – The four pens contain 2, 2, 2, and 6 cows. The key fact is that the blind cowboy needed to ask about a single cow in any pen. There is a second arrangement 1, 3, 4, and 4 which has a matching product, namely 48. All other partitions have unique products.
    8. Grandpa's Birthday – my grandfather was born on December 30, and so for part of that day, he was 52 before turning 53. He spoke to me on New Year's day (in the year he would turn 54), and told me he wold be 55 the following year.
    9. Magic Cube – top layer: 4, 12, 26; 11, 25, 6; 27, 5, 10; middle layer, respectively: 20, 7, 15; 9, 14, 19; 13, 21, 8; bottom layer: 18, 23, 1; 22, 3, 17; 2, 16, 24.
    10. Train tunnel – the problem is solved most easily by drawing three pictures. First, Gus and Freda are 2/5 through the tunnel. We don't know where the train is. Second, since they run in opposite directions at the same speed, one escapes just as the train arrives at that end of the tunnel. The other child has 1/5 of the tunnel remaining to cover in the same time the train travels the entire tunnel. The train is going 75 mph or five times faster than the runners. Try solving for other fractions. Is there always a solution? Is there a formula for the train's speed?
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