Help the poor paranoid scientists!

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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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In response to the post “Stacking Cannonballs” Trevor H. writes:

 I was very intrigued by the recursive sequence you mentioned in the past two episodes–the sequence that begins with 1 and each successive term is the average of all the previous terms times some constant. I have always been fascinated by Pascal’s triangle and all of its surprise appearances in mathematics. Also, my fist encounter with doing mathematics for fun out of my own curiosity was to find a formula for triangle numbers. Like Kyle, I was inspired by bowling pin arrangements. The experience was very rewarding and I have been in love with mathematics ever since.

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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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Pascal’s triangle, with its host of nifty tricks, provides the surprising solution to last weeks’ puzzle on sequences of averages. 

As a bonus puzzle, not mentioned in the podcast, consider the following variation with a completely different solution: Our sequence starts 

1, 1, …

Now each additional term is twice the average of all the earlier terms, not including the terms immediate predecessor! So, the third term is twice the average of 1, i.e. 2. We have now 1,1,2 …

The fourth term is twice the average of 1 & 1, i.e. 2 and we have 1,1,2,2

Continuing in this way we get 1, 1, 2, 2, 8/3, 3, etc.  The sequence wobbles around, but will grow steadily. But the remarkable thing is that the nth term, divided by n, tends to exactly (1 – 1/e^2)/2, a fact well worth trying to prove!

Dirk Huylebrouck, the Mathematical Tourist columnist in the Mathematical Intelligencer, tells us about the remarkable Ishango bone, a 22,000 year old arithmetical exercise!

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In August 2008, the 45th known Mersenne prime, a mere 2^43,112,609-1 was discovered by the Great Internet Mersenne Prime Search! Our puzzle this week is really just to rediscover for yourself proofs that 

• if a number of the form 2^N-1 is prime, then N must also be prime (Or contrapositively, if N is composite, then 2^N-1 is also composite)
• if a number of the form 2^N-1 is prime then the number 2^(N-1) x (2^N-1) is perfect— that is, it is the sum of all its proper divisors.

For example, 2³ – 1 = 7, which happens to be prime. 2² x (2³-1) = 28, which has proper divisors 1, 2, 4, 7, and 14, which sum to (drumroll) 28.

For fun you might look around for numbers of the form 2^{a prime} -1 that are not themselves prime; this shouldn’t take too long since these are far more common than that those that are, the Mersenne Primes.

If you want a little more of a challenge, try to prove that

• any even perfect number must be of this form

and if you want to be really famous, settle the conjectures that

• this takes care of everything—in other words that there are no odd proper numbers
• but that there are in fact infinitely many Mersenne primes and so infinitely many even perfect numbers

Just why does e appear in so many guises?

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We settle some business and address the game of Plinko.

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We discuss math on TV, the smallest ungoogleable number and a devilish game with billiard balls.

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