Bokdol Math (복돌수학)
Bokdol Math (복돌수학)
Free math problems from basic math to algebra, geometry and beyond. Bokdol wants to make math fun for everyone.
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- Meeting Bokdol
You are excited to meet Bokdol for the first time. Unfortunately, your sense of direction is terrible to the point where you move randomly from where you are (← left, → right, ↑ up, ↓ down). How many moves on average will you need to get to Bokdol? One move counts as moving from a box to a box, and you cannot move diagonally or move outside the boxes.
Click here to see the answer!
The answer is 4 moves.
Let's label the boxes first - the box you are at is "z", the one above you (or left to Bokdol) is "x", the one on your right (or under Bokdol) is "y", and the one Bokdol is at is "b". Assume that z', x', y', and b' each represent the number of moves needed on average to get to b from the corresponding box. For example, x' means the number of moves needed on average to get to b from x. What we want to know is z'.
Naturally, b' equals 0 because you need 0 moves to get to b from b. What about x' and y'? x' equals 1 + (1/2)z' + (1/2)b'. 1 means you need to move from x at least once (to either z or b), and there is 1/2 chance that you will get to z and 1/2 chance that you will get to b. As we said before, z' represents the number of moves needed on average to get to b from z, and b' represents the number of moves needed on average to get to b from b (which is 0). So the equation becomes x' = 1 + (1/2)z'. Similarly for y', y' equals 1 + (1/2)z'. We have three equations so far:
b' = 0
x' = 1 + (1/2)z'
y' = 1 + (1/2)z'
Now, the important question is the value of z'. z' equals 1 + (1/2)x' + (1/2)y'. You need to move from z at least once, and there is 1/2 chance that you will get to x and 1/2 chance that you will get to y. Since we already have the equations for x and y that involve z':
z' = 1 + (1/2)x' + (1/2)y'
= 1 + (1/2)(1+(1/2)z') + (1/2)(1+(1/2)z')
...
z'= 4
The value of z' (the number of moves needed on average to get to b from z) is 4.
(Updated Jan 3 2021)
(Updated Jan 3 2021)
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