Why the middle-third rule doesn’t cause instability
Jenga is a family game, which is much enjoyed in our house. The game starts by constructing a wooden tower using layers of three timber bricks. The orientation of the bricks alternates by 90 degrees between layers.
Each player takes a turn to remove one brick from the tower and then to place it on top. This simultaneously adds to the height of the tower while reducing the cross section of the lower portion. This process continues until eventually the tower topples, either during withdrawal of a brick or as it is placed on top. The last person to touch the tower is the loser.
In my experience collapse happens far more frequently while bricks are removed than when they are added. I think this is because the bricks are not exactly the same size and therefore, while some can be removed with considerable ease, others are held tightly by friction. I do not know whether this is an intended jeopardy, but it definitely adds to the game, as you are never quite sure, which bricks will stick until after you are committed.
That said, all things being equal, it is possible to take a brick from either the middle position in any given layer, or from one of the two outer positions, without toppling the tower.
This is interesting, because it undermines a common usage of the so called ‘middle-third’ rule. The middle-third rule essentially states that a structure will avoid tension providing its centre of gravity lies within the middle third of its thickness. Since masonry structures cannot resist tension this is often taken to mean that they will become unstable if the middle third rule is breached, however, as we are about to see, this is not so.
The centre of gravity of a structure is the axis through which its weight acts. For example, the centre of gravity of a uniformly thick wall would be a vertical axis through the middle of the wall. Self-evidently the center of gravity for such a structure falls within the wall’s middle third.
If we apply this logic to a Jenga tower we can remove the middle brick from any layer and neither the centre of gravity nor the middle third will change i.e. the centre of gravity remains in the middle of the tower and therefore the tower remains stable. If, however, we remove a brick from one of the two outer positions something interesting happens.
The centre of gravity above this level remains in the middle of the tower, however at the level with a missing brick the middle third has shifted into the two remaining bricks i.e. it is the middle third of two rather than three bricks.
At this level the middle third therefore extends from 2/9 [1] to 4/9 [2] of the full tower thickness. The centre of gravity of the wall above remains 1/2 of the tower thickness. For ease of comparison we can convert these fractions so that they have the same denominator. The middle third is from 4/18 to 8/18 of the tower thickness, while the centre of gravity is 9/18 i.e. 9/18 sits outside the middle third, which is limited to 8/18.
Since the tower does not collapse the middle third rule, while a cautious limitation, cannot represent the point at which the tower becomes unstable.
The next step would be to modify the rule from the middle third to the middle half. In this scenario the middle half, at the level with two bricks, would extend from 1/6 [3] to 3/6 [4] of the full tower thickness. Of course 3/6 is equivalent to 1/2, which corresponds to the exact position of the tower’s centre of gravity at the level above. This means that the tower is in theory just stable, but ought to topple if it moves even a tiny fraction. Obviously, this can’t be right either or the game would be impossible to play.
Perhaps a good way to think about why this can’t be the correct limit is to imagine what would happen if we removed the remaining outside brick leaving only the centre brick. In this scenario clearly the structure remains balanced and will not topple.
If we now apply the middle third rule to the remaining middle brick the edge of the no-tension zone would extend to 5/9 [5] of the overall tower thickness. This provides a margin of safety of 1/18 relative to the centre of gravity of the tower above. This margin is probably slightly inaccurate, because the tower has not been built under laboratory conditions with perfect alignment between layers. We also know from our earlier discussion that the blocks are not all precisely the same size. This means that in practical terms the tipping point probably occurs if the centre of gravity moves either a little less or a little more than 5/9 of the tower thickness.
If we take the traditional Jenga tower to be 45 mm wide then the margin of safety is circa 2.5 mm. This is perhaps just big enough to tolerate a small disturbance while removing and placing a brick. It is also just small enough to provide a level of jeopardy that makes the game interesting.
It is also worth noting that 5/9 of the full tower width is 25 mm, which equates to 5/6 of the two bricks we started with after removing one of the outer bricks. Thus the 1/3 rule is exceeded by a significant margin.
Of course it becomes easier to breach the margin of safety as the structure becomes taller. This is because the structure becomes top heavy and therefore has greater momentum if disturbed.
[2] 2x 1/3 x 2/3
[3] 1/4 x 2/3
[4] 1/4 x 2/3 + 1/2 x 2/3
[5] 1/3 + 2/3 x 1/3
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