For this homework, you will work on a problem named “who gets the cake”. Let’s start with the introduction of this problem. Imagine that there is a piece of cake and several people want it. They decide who can have it by playing a game: They form a circle and choose an integer k greater than one. They count 1, 2, 3, …, k. The k-th person is eliminated. They keep counting until only one person is left. This person who is left gets the cake.

Please notice that there are different definitions of this problem. *Your solution must follow the definition here.* More precisely, this is how the method works: There are n people (n is an integer), represented by n elements in an array. The elements are counted as 1, 2, 3, … When the value k is counted, this element is removed in future counting and the next element starts as 1 again. When reaching the end of the array, the counting *wrap around* to the beginning of the array (skipping the elements that have already been eliminated). Please notice that in C arrays, indexes *always* start at zero but in this problem counting starts at one. Both n and k have to be greater than one. *It is possible that k is greater than n.*

Arrays are sequences of data types laid out next to each other in memory. C treats array data types specially, allowing you to access them by *indexing* from a base expression:

`int a[10]; //10 integers next to each other`

a[0] = 5; //access the first integer

a[9] = 10; //access the 10th integer

Remember that the first element of an array is at index `0`

, and the last element of an array is at index `size - 1`

.

Note that if you don’t know the size of the array ahead of time, you can *dynamically allocate* the array:

`int * a = malloc(20 * sizeof(* a)); //allocate an array of 20 integers`

a[4] = 20; //access the fifth integer

Do not worry if the syntax of the first line looks mysterious. We will explain it in detail when we discuss pointers and memory allocation. For now, just rely on this making an array of 20 integers!

- The following is an example when the array has 6 elements (n is 6) and k is 3. The eliminated elements in each round are mared by
`X`

. The elements eliminated earlier are marked by`Y`

.

array index012345count12X12Xarray index012345count12YX1Yarray index012345count2XYY1Yarray index012345count2YYYXY

The element of index 0 is left.

- This is the second example. The array has 4 elements (n is 4) and k is 6. This is an example where k is greater than n.

array index0123count1234array index0123count5X12array index0123count3Y45array index0123countXY12array index0123countYY34array index0123countYY5X

The element of index 2 is left.

- This is the third example. The array has 25 elements (n is 25) and k is 7.

array index0123456789101112131415161718192021222324count123456X123456X123456X1234array index0123456789101112131415161718192021222324count56X123Y456X12Y3456X1Y2345array index0123456789101112131415161718192021222324count6XY123Y456YX1Y2345Y6YX123array index0123456789101112131415161718192021222324count4YY56XY123YY4Y56X1Y2YY345array index0123456789101112131415161718192021222324count6YYX1YY234YY5Y6XY1Y2YY345array index0123456789101112131415161718192021222324count6YYYXYY123YY4Y5YY6YXYY123array index0123456789101112131415161718192021222324count4YYYYYY56XYY1Y2YY3YYYY456array index0123456789101112131415161718192021222324countXYYYYYY12YYY3Y4YY5YYYY6X1array index0123456789101112131415161718192021222324countYYYYYYY23YYY4Y5YY6YYYYXY1array index0123456789101112131415161718192021222324countYYYYYYY23YYY4Y5YY6YYYYYYXarray index0123456789101112131415161718192021222324countYYYYYYY12YYY3Y4YY5YYYYYYYarray index0123456789101112131415161718192021222324countYYYYYYY6XYYY1Y2YY3YYYYYYYarray index0123456789101112131415161718192021222324countYYYYYYY4YYYY5Y6YYXYYYYYYYarray index0123456789101112131415161718192021222324countYYYYYYY1YYYY2Y3YYYYYYYYYYarray index0123456789101112131415161718192021222324countYYYYYYY4YYYY5Y6YYYYYYYYYYarray index0123456789101112131415161718192021222324countYYYYYYYXYYYY1Y2YYYYYYYYYYarray index0123456789101112131415161718192021222324countYYYYYYYYYYYY3Y4YYYYYYYYYYarray index0123456789101112131415161718192021222324countYYYYYYYYYYYY5Y6YYYYYYYYYYarray index0123456789101112131415161718192021222324countYYYYYYYYYYYYXY6YYYYYYYYYY

The element of index 14 is left.

The table uses `X`

and `Y`

for clarity. Your program should use only `X`

.

Write the `eliminate`

function in `eliminate.c`

to print the index of the eliminated elements *in order*.

In the first example (n=6, k=3), the output is

`2`

5

3

1

4

0

In the second example (n=4, k=6), the output is

`1`

0

3

2

In the third example (n=25, k=7), the output is

`6`

13

20

2

10

18

1

11

21

5

16

3

15

4

19

9

0

23

22

24

8

17

7

12

14

The function is called `eliminate`

, not `select`

because `select`

is a C function for communication. If you want to know the definition of the `select`

function, search `Linux manual select`

.

Testing: You are given the expected output for the three examples discussed above. Feel free to create more to test your program.

In the provided Makefile, you will use the `diff`

command to compare between your output and the expected output.

The `diff`

command: `diff $YOUR_OUTPUT $EXPECTED_OUTPUT`

displays the differences between the two files line-by-line. When the two files are identical, `diff`

will be silent.

You will receive zero if your program has error or warning from `gcc`

.

The grading will be based on the test cases run by the teaching staffs. Your score is proportional to the number of test cases that your program passes. Passing means the program returns `EXIT_SUCCESS`

and provides correct outputs that match the expected outputs.

A mathematical question is to determine which element is left *without* counting 1, 2, … If you are interested in this topic, please read the book Concrete Mathematics by Ronald L. Graham, Donald E. Knuth, and Oren Patashnik.

Is this a real problem? Is there any real application? Yes. In distributed systems (such as the Internet), sometimes different machines need to agree on something. For example, a group of machines want to find one representative for external communication.

This problem is inspired by the “Josephus problem”. History (based on Wikipedia): Flavius Josephus and 40 soldiers were trapped in a cave by Roman soldiers. They chose suicide over capture, and decided the order is determined by the following method: they form a circle and set an integer k greater than one. Then, the group starts with 1, 2, … The person that counts k is eliminated. The process continues until all are eliminated. The question is where Josephus should stand at the beginning so that he is the last remaining person.

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