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A Child and a Staircase


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Either way, hh3 got it wrong with his answer. There was not one way as explained by him, but 2 ways.

dude... look at the spoiler in my second post... hi hi.. but the answer still could me wrong..bt anyways..
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I would say 89 because

1 1 1 1 1 1 1 1 1 1 (only 1 combination)

1 1 1 1 1 1 1 1 2 (9 combinations)

1 1 1 1 1 1 2 2 (28 combinations)

1 1 1 1 2 2 2 (35 combinations)

1 1 2 2 2 2 (15 combinations)

2 2 2 2 2 (only 1 combination)

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Let F(n) be the number of different ways staircase of n steps can be climbed.

You can start either with 2 steps (and n-2 remaining) or with 1 step (and n-1 remaining),

which leads to a recurrence relation:

F(n) = F(n-1) + F(n-2)

Obviously

F(0) = 1, F(1),

and

F(n+2) = F(n) + F(n+1)

so you are looking for 10th Fibonacci number, which is 89.

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Unless I missed something...

All 1 steps - 1 way

One 2 step - 8 ways

Two 2 steps - 6 ways

Three 2 steps - 4 ways

Four 2 steps - 2 ways

Five 2 steps - 1 way

Total = 22 ways

Not only did I miss something, I missed a lot - combinations of the 2 step options! The actual ways would be much more.
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442.

Since we can solve this with either Fibonacii numbers or combinations, we can conclude that

F10 = C(10,0) + C(9.1)+ ...+c(5.5).

More generally Fn = C(n,0) + C(n-1.1) + C(n-2,2) + ... as long as the first number is no les than the second.

NOTE If you interpet the question to mean 10 steps plus the upper landing the answer would be F11, a point one poster raised. In addition if we distinquish between first strride right foot and first stride left we would double our answer.

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Solved. Congrats to all of you who got it correct.

Yes, 89 is the correct answer. i.e. 10thnumber in the Fibonacci series (1,2,3,5,8,13,21,34,55,89,144).

However, I agrees with Rackmond question's too, if we consider 10 steps + a upper landing then the answer will be 144.

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I think 89 is correct. Here's a simple C# implementation that you can test out if you want to use LinqPad:

void Main()
{
countWays(10).Dump();
}
int countWays(int steps)
{
if (steps < 0)
return 0;
if (steps == 0)
return 1;
if (_map.ContainsKey(steps))
return _map[steps];
_map[steps] = countWays(steps - 1) + countWays(steps - 2);
return _map[steps];
}
private readonly Dictionary<int, int> _map = new Dictionary<int, int>();
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