Lara starts at the bottom of a long staircase. She climbs exactly 2/3 of the stairs. Then, she goes back down exactly 1/2 of the way to the bottom. From that spot, she climbs exactly 2/3 of the way to the top. Finally from there, she climbs 6 stairs to reach the top. How many stairs are in the staircase?

Ah, the infamous staircase riddle! Let me put on my mathematician's wig and help you out.

Let's assume there are "X" stairs in the staircase. Lara climbs 2/3 of the stairs, so she reaches the 2/3X mark. Then, she descends back down to the 1/2X mark. From there, she climbs 2/3 of the way to the top, which means she reaches the (1/2X + 2/3X) mark.

Now, to determine the total number of stairs, we know that Lara climbs 6 stairs from the (1/2X + 2/3X) mark to the top, which means 6 stairs equals 1/3X.

Now we can find the value of X by solving the equation:

1/3X = 6

Multiplying both sides by 3, we get:

X = 6 x 3

X = 18

So, the staircase has 18 stairs!

Now, if only Lara could stop going up and down stairs and settle for an elevator. It would save her a lot of time and confusion!

Assume that there are x steps in all. Lara goes up (2x/3) steps and returns half the number of steps taken to 2x/6 or to (x/3) up from start.

After this there are (2x/3) steps still to go of which she once again takes (2/3)(2x/3)=(4x/9) steps to reach a spot which 6 steps away from the top.
Hence, x/3 + 4x/9 + 6 = x or x=27 steps.

Let's solve this step-by-step.

Step 1: Climbing 2/3 of the stairs.
Let's say there are x stairs in the staircase.
Lara climbs 2/3 of x stairs, so she climbs (2/3)x stairs.

Step 2: Going back down 1/2 of the way to the bottom.
From the spot where Lara went back down, she has climbed (2/3)x stairs.
She goes back down 1/2 of the way, which means she descends (1/2)(2/3)x = (1/3)x stairs.

Step 3: Climbing 2/3 of the way to the top.
After descending (1/3)x stairs, Lara is at (2/3)x - (1/3)x = (1/3)x stairs from the bottom.
From there, she climbs 2/3 of the remaining distance to the top, which is (2/3)((1/3)x) = (2/9)x stairs.

Step 4: Climbing the final 6 stairs.
After climbing (2/3)x and descending (1/3)x stairs, Lara is at (1/3)x + (2/9)x = (7/9)x stairs from the bottom.
Finally, she climbs the remaining 6 stairs to reach the top.
Therefore, (7/9)x + 6 = x, as Lara reaches the top.

Let's solve this equation for x:

(7/9)x + 6 = x
7x + 54 = 9x
54 = 2x
x = 27

So, there are 27 stairs in the staircase.

To determine the total number of stairs in the staircase, we need to analyze the information provided step by step.

First, Lara climbs exactly 2/3 of the stairs. Let's assume the total number of stairs is represented by 'x'. Therefore, the number of stairs Lara climbs in this step is (2/3)x.

Next, Lara goes back down exactly 1/2 of the way to the bottom. This means she descends (1/2) * (2/3)x = (1/3)x stairs.

From that spot, Lara climbs exactly 2/3 of the way to the top. Since she is already (1/3)x stairs from the top, the number of stairs she climbs in this step is (2/3) * (1/3)x = (2/9)x.

Finally, Lara climbs 6 stairs to reach the top from that point. We can set up an equation to represent the total number of stairs:

(2/3)x - (1/3)x + (2/9)x + 6 = x

To solve this equation, we need to eliminate the fractions. We can do this by multiplying through by the least common multiple (LCM) of the denominators, which is 9:

9 * [(2/3)x] - 9 * [(1/3)x] + 9 * [(2/9)x] + 9 * [6] = 9 * [x]

Simplifying this equation:

6x - 3x + 2x + 54 = 9x

Combining like terms:

5x + 54 = 9x

Subtracting 5x from both sides:

54 = 4x

Dividing by 4:

x = 13.5

Since the total number of stairs must be a whole number, we round up to the nearest whole number because Lara cannot have climbed half a stair:

x = 14

Therefore, there are 14 stairs in the staircase.