Problem of the Week 7

There is  a limited number of yearly calendars.  How many different possible calendars are there?

How many years does it take before each calendar is used at least once?  Start counting with the year 2024.

Alternate Problem 7

A set containing one element can be arranged in only one way.  A set containing two elements can be arranged in two different ways: (a,b) and (b,a) for example.  In how many ways can a set containing three elements be arranged?  What about a set containing four elements?

Complete the following table, where n stands for the number of elements and a stands for the number of arrangements.

 n a Pattern 1 1 2 2 1 x 2 3 6 4 5 6 n

Extension Problem 7

Suppose 19 people are arranged around a circle and numbered from 1 through 19.  Starting with 1, eliminate every second person.  Thus, for 19 people, the elimination is 2, 4, 6, 8, 10, 12, 14, 16, 18, 1, 5, 9, 13, 17, 3, 11, 19, 15.  The remaining person is person number 7.

a.  For values of n from 2 through 20, make a table showing the remaining number after the process of elimination.

b.  Formulate a conjecture about which values of n have 1 for the remaining number.

c.  On the basis f the conjecture in b and the pattern that appears, find the remaining number if n = 300.