Can You Win Riddler Jenga?

Riddler Classic

In the game of Jenga, you build a tower and then remove its blocks, one at a time, until the tower collapses. But in Riddler Jenga, you start with one block and then place more blocks on top of it, one at a time.

All the blocks have the same alignment (e.g., east-west). Importantly, whenever you place a block, its center is picked randomly along the block directly beneath it. For example, the following animation shows Riddler Jenga towers that were randomly constructed before ultimately collapsing when the fifth, 10th and 15th blocks were placed. The block highlighted in red is the one above which the blocks were no longer balanced.

On average, how many blocks must you place so that your tower collapses — that is, until at least one block falls off?

(Note: This problem is *not asking for the average height of the tower after any unbalanced blocks have fallen off. It is asking for the average number of blocks added in order to make the tower collapse in the first place.)

Solution

If the average center of $k$ blocks above a block are beyond the horizontal dimensions of that block, then all $k$ blocks will fall off.

Assume a stack of $n$ blocks which are stable. The addition of block $n+1$ will cause a collapse of $k$ blocks, for $2 \leq k \leq n$, if the average center of blocks $n-k+2$ to $n+1$ is beyond the horizontal dimensions of block $n-k+1$.

$k$ cannot be $1$ because the center of the top block is within the bounds of the previous block, and it cannot be all $n+1$ because the bottom block is stable.

I ran 100,000,000 game simulations and tallied the results. The frequency distribution and cumulative percent are shown below.

Answer

Using values from the data above, the average height of a tower before any part of it being unbalanced and falling off is

$$\approx 7.1127$$

Rohan Lewis

2021.02.20

Code can be found here.