1. 25 people are sitting in 3 concentric circles with one person sitting
at the center. Every circle has 8 people sitting, and everyone is
sitting on radii that are spaced 45 degrees, with 4 people on every such
line. Find out number of permutations in such a scenario.
2. There is a circle, for which a maximal square is inscribed in it. After erasing two opposite sides of a square, places were marked for people to sit on the sides of square and on the circle. Circle has 16 people equi-spaced, and each side has 3 people equi-spaced (There is one person at the ends of each side, where it meets the circle). Given 22 people, how many such arrangements are possible?
3. Consider a rectangle of dimensions m * n. There are 3 possible moves, for someone at square (0, 0) in order to reach square (m, n). Upwards, which moves one from (a, b) to (a, b+1), right side which is (a, b) to (a+1, b). and diagonal, which is (a, b) to (a+1, b+1). How many possible routes are there to reach (m, n) from (0, 0).
4. Consider a move where; instead of diagonal move, one moves to (a+2, b+1) from (a, b) in two moves. Calculate the number of different routes, and minimal number of moves.
5. Generalize for the case of (a+n, b+1) from (a, b) in n moves, and minimal number of moves.
6. Generalize for the case of (a+1, b+n) from (a, b) in n moves, and minimal number of moves.
7. Generalize for the case of (a+n, b+1) from (a, b) in 1 move, and minimal number of moves.
8. Generalize for the case of (a+1, b+n) from (a, b) in 1 move, and minimal number of moves.
9. Consider the circular configuration in problem 1. From any point (point is where a person used to sit in problem 1) on the outermost circle, reaching the center can be done by moving from one point to another by
a. going inward
b. going to left
c. going to right
Calculate the number of possible routes. One is allowed to visit a point, at-most once only.
10. Generalize for n points around the circle's circumference.
11. Generalize for n concentric circles.
12. Generalize for both as in problems 10 and 11
13. Do problem 9, without "move b".
14. Consider a circular sector of angle 90 degrees. Consider n such concentric sectors. The various sectors are joined by straight lines (radii) at angle 0, 10, 20, ... 90 degrees. Considering points at every intersection, and using moves a,b and c from 9, find how many possible paths exist to reach the center from any point on the outermost sector.
15. A hop is similar to diagonal move, in sense that, it is used to jump when there can be a diagonal between points. Include left side inward hop as a possible move, and solve 9.
16. Include hop move and solve problem 14.
17. Include hops like what is presented in problems 5-8 and solve 9.
18. Include hops like what is presented in problems 5-8 and solve 14.
19. Consider natural numbers in a line, starting from zero. 2 pieces move along the line as follows, both starting from zero.
i. Roll a dice, and move accordingly, so many points, from 1 to 6.
Find out the number of possible locations (as ordered pairs, first piece followed by the second) after n moves.
20. Consider the following change to 19.
i. If piece 1 plays dice and lands up in the same square as piece 2, it will be reset, meaning sent back to 0 on the line. (Similarly for piece 2).
Now find the number of possible locations, after say 100 moves.
21. What is the expected position of piece (any piece) from question 19, after n moves.
22. What is the expected position of piece from question 20, after n moves.
23. Consider the following:
i. If a roll of dice results in a 6, the piece is allowed another roll. It finally moves the sum of rolled dice, as in, first comes 6. Next comes 5 in the rolls. Then it moves 6 + 5 = 11 points on the line.
Now, find the number of possible locations.
24. Use 23, and repeat with condition of question 20
25. Use 23, and find the expected position of a piece (Use conditions from q19).
26. Use 23, and find the expected position of a piece (Use conditions from q20).
27. What is the expected distance between the pieces, from q19. (There is no negative distance, all distances are positive).
28. Repeat 27, for q20 conditions
29. Repeat 27, for q23 conditions
30. Find the expected location of a piece, after the first move, under following conditions on top of q19 conditions.
i. If a roll results in a 6, the dice are allowed to be rolled again. If it gets another 6, it is rolled once again and so on and so forth.
2. There is a circle, for which a maximal square is inscribed in it. After erasing two opposite sides of a square, places were marked for people to sit on the sides of square and on the circle. Circle has 16 people equi-spaced, and each side has 3 people equi-spaced (There is one person at the ends of each side, where it meets the circle). Given 22 people, how many such arrangements are possible?
3. Consider a rectangle of dimensions m * n. There are 3 possible moves, for someone at square (0, 0) in order to reach square (m, n). Upwards, which moves one from (a, b) to (a, b+1), right side which is (a, b) to (a+1, b). and diagonal, which is (a, b) to (a+1, b+1). How many possible routes are there to reach (m, n) from (0, 0).
4. Consider a move where; instead of diagonal move, one moves to (a+2, b+1) from (a, b) in two moves. Calculate the number of different routes, and minimal number of moves.
5. Generalize for the case of (a+n, b+1) from (a, b) in n moves, and minimal number of moves.
6. Generalize for the case of (a+1, b+n) from (a, b) in n moves, and minimal number of moves.
7. Generalize for the case of (a+n, b+1) from (a, b) in 1 move, and minimal number of moves.
8. Generalize for the case of (a+1, b+n) from (a, b) in 1 move, and minimal number of moves.
9. Consider the circular configuration in problem 1. From any point (point is where a person used to sit in problem 1) on the outermost circle, reaching the center can be done by moving from one point to another by
a. going inward
b. going to left
c. going to right
Calculate the number of possible routes. One is allowed to visit a point, at-most once only.
10. Generalize for n points around the circle's circumference.
11. Generalize for n concentric circles.
12. Generalize for both as in problems 10 and 11
13. Do problem 9, without "move b".
14. Consider a circular sector of angle 90 degrees. Consider n such concentric sectors. The various sectors are joined by straight lines (radii) at angle 0, 10, 20, ... 90 degrees. Considering points at every intersection, and using moves a,b and c from 9, find how many possible paths exist to reach the center from any point on the outermost sector.
15. A hop is similar to diagonal move, in sense that, it is used to jump when there can be a diagonal between points. Include left side inward hop as a possible move, and solve 9.
16. Include hop move and solve problem 14.
17. Include hops like what is presented in problems 5-8 and solve 9.
18. Include hops like what is presented in problems 5-8 and solve 14.
19. Consider natural numbers in a line, starting from zero. 2 pieces move along the line as follows, both starting from zero.
i. Roll a dice, and move accordingly, so many points, from 1 to 6.
Find out the number of possible locations (as ordered pairs, first piece followed by the second) after n moves.
20. Consider the following change to 19.
i. If piece 1 plays dice and lands up in the same square as piece 2, it will be reset, meaning sent back to 0 on the line. (Similarly for piece 2).
Now find the number of possible locations, after say 100 moves.
21. What is the expected position of piece (any piece) from question 19, after n moves.
22. What is the expected position of piece from question 20, after n moves.
23. Consider the following:
i. If a roll of dice results in a 6, the piece is allowed another roll. It finally moves the sum of rolled dice, as in, first comes 6. Next comes 5 in the rolls. Then it moves 6 + 5 = 11 points on the line.
Now, find the number of possible locations.
24. Use 23, and repeat with condition of question 20
25. Use 23, and find the expected position of a piece (Use conditions from q19).
26. Use 23, and find the expected position of a piece (Use conditions from q20).
27. What is the expected distance between the pieces, from q19. (There is no negative distance, all distances are positive).
28. Repeat 27, for q20 conditions
29. Repeat 27, for q23 conditions
30. Find the expected location of a piece, after the first move, under following conditions on top of q19 conditions.
i. If a roll results in a 6, the dice are allowed to be rolled again. If it gets another 6, it is rolled once again and so on and so forth.
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