2.9 The Problems Page
Chapter
Chapter 2
Section
2.9
11mathHarcourt2.9 12 Videos

Recall the problem from Chapter 1 in which you New Tab ed to move blue and green discs so that the positioning of the discs was reversed. Here is a variation of the game. What's different? Two things. First, we have three blue and three green discs, but also a one-disc space between them. Second, there are two allowable moves; you can move a disc one position if there is a blank space beside it, and you can jump one disc over another if there is an empty space to jump into. What is the minimum number of moves required to get the green discs to the three positions on the left and the blue discs to the three positions on the right? Can you give a general solution for \displaystyle k  discs of each colour?

Q1

This is a set of questions about prime numbers. Your task is to answer the question posed and to draw conclusions where you can.

The numbers 2 and 3 differ by one. Can you find another such pair of primes?

Q2a

This is a set of questions about prime numbers. Your task is to answer the question posed and to draw conclusions where you can.

The numbers 3 and 5 differ by two. Can you find another such pair of primes? How many?

Q2b

This is a set of questions about prime numbers. Your task is to answer the question posed and to draw conclusions where you can. The numbers 3,5, and 7 are a triple set of primes differing by two. Can you find another such triple?

Q2c

This is a set of questions about prime numbers. Your task is to answer the question posed and to draw conclusions where you can. Can you find a prime that is one less than a perfect square? Can you find more than one such prime?

Q2d

This is a set of questions about prime numbers. Your task is to answer the question posed and to draw conclusions where you can. Can you find a prime that is one more than a perfect square? Can you find more than one such prime?

Q2e

When \displaystyle f(x)  is divided by \displaystyle 3 x+1 , a quotient of \displaystyle 2 x-3  and a remainder of 5 are obtained. What is \displaystyle f(x)  ?

Q3

Bus A and bus B leave the same terminal at 09:00. Bus A requires 45 min to complete its route while bus \displaystyle \mathrm{B}  requires 54 min for its route. If the buses continue running, what is the next time the two leave the terminal together?

Q4

If two poles \displaystyle 10 \mathrm{~m}  and \displaystyle 15 \mathrm{~m}  high are \displaystyle 25 \mathrm{~m}  apart, what is the height of the point of intersection of the lines that run from the top of each pole to the foot of the other pole?

We define a sequence of numbers by using the symbol \displaystyle t_{k}  where \displaystyle k=1,2,3,4, \ldots  This means that \displaystyle t_{1}  represents the first number in the sequence, \displaystyle t_{2}  the second, and so on. If \displaystyle t_{1}=1, t_{2}=2 , and every number thereafter is obtained from the formula \displaystyle t_{k+1}=\frac{t_{k}+1}{t_{k-1}}\left(\right.  for example, \displaystyle t_{3}=\frac{t_{2}+1}{t_{1}}=\frac{2+1}{1}=3  ), what is \displaystyle t_{63}  ? Can you determine the sum of the first 100 numbers in the sequence?
Show that for the set of numbers in Problem \displaystyle 6, t_{6}=t_{1}  and \displaystyle t_{7}=t_{2} .