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MathMagic Cycle 16: Level 7-9 Regular

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Palindromes (Quick, look it up in the math dictionary) Definition:

One of the most famous palindromes was about a US president:
          A man, a plan, a canal - Panama!             Who:

A number such as 1331 is also a palindrome. It has "bilateral symmetry".
If "bi" means two, what do you think that concept means? Does 1356531
have bilateral symmetry? Explain.

A famous unsolved number problem called the "palindrome conjecture" says
that you can start with any number greater than 10, reverse it, and add
the two numbers. After a certain number of times repeating this process,
you will come up with a palindrome:

1) 86+68=154  2) 154+451= 605  3) 605+506= 1111 which is a palindrome.

Experiment with say five numbers. Try to stay in the range of 10 to 1000.
How many steps did you have to do before a palindrome came up?

Try 196. What appears to happen?

By using a few examples, can you show that the statement "Every palin-
dromic number with an even number of digits is a multiple of 11" is in
fact a true statement?

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MathMagic Cycle 16: Level 7-9 Advanced

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Lewis Carroll (Alice in Mathland?) was not only a very talented writer,
but he was quite fond of mathematical puzzles. The following is attributed
to him:

     "A queen and her son and daughter are being held captive in the top
room of a high tower. Outside their window is a pulley with a rope over
it, and a basket at each end of the rope. The baskets are of equal weight.
The one outside the window is empty, and the other on the ground contains
a stone with a mass of 30 kilograms. The stone serves as a counterweight.

     There is enough friction in the pulley so that it is safe for anyone
to be lowered in one basket provided his or her mass is not greater than
the mass of the other basket by more than 6 kilograms. If the difference
is greater than six kilograms, they come down with such speed that the
bump at the bottom might injure them. Of course, when one basket goes down
the other basket goes up to the window.

     The queen's mass is 78 kilograms, the daughter's is 42 kilograms, and
the son's is 36 kilograms. What is the simplest algorithm (fewest number
of steps) by which they can all get safely to the ground? The basket is
large enough to hold any two people, or one person and the stone. No one
assists the prisoners in escaping, nor can they help themselves by pulling
on the rope. In other words, the pulley operates only when the mass in one
basket exceeds the mass in the other."

Assuming the last person gets out of the way quickly, make a diagram of
the movements (____down; _____up) - What is the algorithm Carroll is
after?

(From Aha! Insight by Martin Gardner)
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