HW5comp

Ex S2:

There are three rods (rod A, rod B, and rod C) and 20 disks of different sizes, which can slide onto any rod. The 20 disks are placed in ascending order of size on rod A, while the smallest at the top, and the largest at the bottom.

Question: Please calculate the minimum number of steps required to move the 20 disks from rod A to rod C, obeying the following rules: 1. Only one disk can be moved in a step. 2. No larger disk can be placed on top of a smaller disk. 3. Please answer this question with a recursive function.

hanoi_tower <- function(n) {
        # by the pattern, the largest disc will be move with 1 step, the rest are in 2 steps
        if (n == 1) {
                cat("move disc(1)\n")
                return(1)
        } else {
                cat("move disc(", n, ")\n", sep = "")
                return(1+2*hanoi_tower(n-1)) -0.5
        }
}

hanoi_tower(20)

## move disc(20)
## move disc(19)
## move disc(18)
## move disc(17)
## move disc(16)
## move disc(15)
## move disc(14)
## move disc(13)
## move disc(12)
## move disc(11)
## move disc(10)
## move disc(9)
## move disc(8)
## move disc(7)
## move disc(6)
## move disc(5)
## move disc(4)
## move disc(3)
## move disc(2)
## move disc(1)

## [1] 1048575

Ex S3:

Please use the “apply” command to calculate the medians, maximums, and minimums, of each row and each column. Missing values could be ignored.

matrix <- matrix(
        c(3600, 5000, 12000, NA, 1000, 2000, 600, 7500, 1800, 9000,
          3600, 4500, 10000, 8500, 3000, 10000, 1000, NA, 1200, 10000,
          3800, 5500, 9000, 6000, 6600, 3000, 9600, 6500, 8200, 8000,
          5000, 6600, 13000, 4500, 5000, NA, 10600, 9500, 7600, 6000,
          6600, 8000, 17000, 3000, 7000, 1000, 12600, 8500, 6000, NA),
        5,10, byrow = TRUE)  

# by row
apply(matrix,1,median, na.rm = T)

## [1] 3600 4500 6550 6600 7000

apply(matrix,1,max, na.rm = T)

## [1] 12000 10000  9600 13000 17000

apply(matrix,1,min, na.rm = T)

## [1]  600 1000 3000 4500 1000

# by col
apply(matrix,2,median, na.rm = T)

##  [1]  3800  5500 12000  5250  5000  2500  9600  8000  6000  8500

apply(matrix,2,max, na.rm = T)

##  [1]  6600  8000 17000  8500  7000 10000 12600  9500  8200 10000

apply(matrix,2,min, na.rm = T)

##  [1] 3600 4500 9000 3000 1000 1000  600 6500 1200 6000

Ex S8

Please find x, y, and z to satisfy Please first build a matrix and use the “solve” command.

x <- matrix(c(1,-3,1,1,-2,3,1,-1,1),3,3,byrow = T)
y <- c(4,6,4)
(sol <- solve(x,y))

## [1] 3 0 1

# x %*% sol