Help the poor paranoid scientists!

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A quite elementary question:

Imagine a tight band wrapped around the Earth (a perfectly spherical Earth!). If one foot is added to the band, it will be possible to lift it uniformly up, away from the surface of the Earth. Will the resulting gap be enough to pass a baseball card under? A baseball? A baseball player? 

Another variation, which is really quite amazing, is what if a foot is added and the band is lifted up in just one spot? How high will the band lift up? Higher than a seball? A baseball player? A baseball stadium?

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Another nice standard; it seems there is not enough information to solve this puzzle but it has a simple solution.

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This is a beautiful puzzle that appears in many different forms.

Arp and Bif are playing with a line of 100 flowers. Each flower is originally open. When an open flower is touched it closes, and when a closed flower is touched, it opens. First they touch every flower in the line, then they touch every other flower in the line, then they touch every third flower, etc. 

When done, which flowers are open, which flowers are closed?

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Edmund Harriss, sometime contributor to the Math Factor, makes his first appearance in this early segment, from February 29, 2004.

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How many ways can the astronauts link up to the space station?

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Given a difference table, as we considered back in EV. What’s the Difference , how do we come up with a polynomial that gives the values on the top row?

For example, suppose we have

-1     -1     3     35     143     399     899 . . . . .
      0     4     32    108     256     500  . . . . .
         4    28    76     148      244  . . . . .
             24    48     72       96   . . . . .
                  24     24    24    . . . . .

What is the polynomial P(n), of degree four, that gives

P(0) = -1 P(1) = -1 P(2) = 3 P(3) = 35 P(4) = 143 , etc.

Can this be expressed simply in terms of the leading values on the left of the table: -1, 0, 4, 24, 24?

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We settle some business and address the game of Plinko.

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We answer last weeks puzzle, discuss the law of small numbers and ask again what is the smallest positive counting number that Google can’t find?

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We conclude our interview with Dana Richards, editor of Martin Gardner’s Colossal Book of Short Puzzles and Problems pondering how to cut a hole through a cube large enough that another, same-sized cube can pass through!

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