Plus An Obstruction (Corrected)

Probability Level 3

Harry lives in a 9 × 9 9\times 9 grid, where his home is in top left corner and office in bottom right corner. He wants to go to his office to work, and then come back home without visiting a cell twice. While going to office, he is only allowed to move one cell right, one cell down, or one cell diagonally below to the right:

While going home, he is only allowed to move one cell left, one cell up , or one cell diagonally above, towards left:

Also, Harry can't pass through the cells marked red on the topmost grid. In how many ways can Harry go to office and come back home ?


The answer is 31752.

Mohit Kumar Basak by Mohit Kumar Basak , 25, Switzerland

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2 solutions

Jeremy Galvagni
Oct 7, 2018

There are 126 126 ways to get from the top center to the right center.

By symmetry, there are also 126 126 ways to get from the left center to bottom center.

Giving a total of 126 × 2 = 252 126 \times 2 = 252 ways to get to work.

Since only the main path not chosen to get to work is available for the return home, there are only 126 126 ways to get home.

Grand total: 252 × 126 = 31752 252 \times 126 = \boxed{31752}

4 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice explanation, I used the exactly same method, I got 63504 which is twice the value you got simply because I forgot that "and then come back home without visiting a cell twice".

Vlad Alexandru - 2 years, 8 months ago

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I often miss those little details as well. Fortunately here you get 3 tries so you can refine your first number.

Jeremy Galvagni - 2 years, 8 months ago

Doh... I did the same mistake... :-/

Geoff Pilling - 2 years, 8 months ago

Let's construct a 4*4 unobstructed grid and find all the ways to go from top left corner to bottom right corner. It is 63.

Now, lets consider the top right 4*4 grid:

There are two ways to enter the grid from the top bottleneck coloured yellow:-

1) Right

2) Diagonal

Similarly, there are two ways to leave the grid to the cell coloured green:-

1) Down from the top cell

2) Diagonal from the top left cell.

When Harry moves right from yellow cell and then reaches the green cell from the cell just above it, he moves across a 4*4 unobstructed grid and has 63 moves.

When Harry moves diagonally from yellow cell and then reaches the green cell from the cell just above it, he moves across a 3*4 unobstructed grid and has 25 moves.

When Harry moves right from yellow cell and then reaches the green cell from the cell diagonally top-left, he moves across a 4*3 unobstructed grid and has 25 moves.

When Harry moves diagonally from yellow cell and then reaches the green cell from the cell diagonally top-left, he moves across a 3*3 unobstructed grid and has 13 moves.

Hence, total ways to enter and leave the top-right 4 * 4 grid = 63 + 25 + 25 +13 = 126

Ways to reach office through top-right 4 * 4 grid and return home = 126*126

Also, Harry can reach office though bottom left 4*4 grid and return home.

Total ways = 126 * 126 * 2 = 15876 * 2 = 31752

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Hi, if you have a better editorial, after solving the problem, you can post solutions.

Mohit Kumar Basak - 2 years, 8 months ago

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