The Blue Eyes Problem and Solution

The problem is this (Source: Terence Tao blog):

There is an island upon which a tribe resides. The tribe consists of 1000 people, with various eye colours. Yet, their religion forbids them to know their own eye color, or even to discuss the topic; thus, each resident can (and does) see the eye colors of all other residents, but has no way of discovering his or her own (there are no reflective surfaces). If a tribesperson does discover [that] his or her own eye color [is blue], then their religion compels them to commit ritual suicide at noon the following day in the village square for all to witness. All the tribespeople are highly logical and devout, and they all know that each other is also highly logical and devout (and they all know that they all know that each other is highly logical and devout, and so forth).

[Added, Feb 15: for the purposes of this logic puzzle, “highly logical” means that any conclusion that can logically deduced from the information and observations available to an islander, will automatically be known to that islander.]

Of the 1000 islanders, it turns out that 100 of them have blue eyes and 900 of them have brown eyes, although the islanders are not initially aware of these statistics (each of them can of course only see 999 of the 1000 tribespeople). [irrelevant]

One day, a blue-eyed foreigner visits to the island and wins the complete trust of the tribe.

One evening, he addresses the entire tribe to thank them for their hospitality.

However, not knowing the customs, the foreigner makes the mistake of mentioning eye color in his address, remarking “how unusual it is to see another blue-eyed person like myself in this region of the world”.

What effect, if anything, does this faux pas have on the tribe?

[bolded] is my edit, edited so my explanation jives with the brain teaser.

The answer is that all the islanders with blue eyes die. Here’s why.

Let n be the number of people with blue eyes. Consider the case n=1. The guy with blue eyes sees nobody else with blue eyes, and so kills himself on the first day (“Day 1”) after the foreigner makes the announcement.

Now consider n=2. Each person with blue eyes sees another person with blue eyes. Let’s use the perspective of Person 2. Person 2 sees one person with blue eyes, Person 1, so he expects Person 1 to kill himself on Day 1. But Person 1 doesn’t, because by symmetry, Person 1 sees the same thing as Person 2. Thus by the fact that Person 1 is alive on Day 1, Person 2 realizes Person 1 sees a person with blue eyes. As there are no other people with blue eyes, Person 2 realizes he must be it. Again by symmetry, Person 1 makes this realization as well. Both kill themselves on Day 2.

n=3 becomes a little complicated. Person 3, let’s say, now sees Person 1 and 2 with blue eyes. Nothing happens on Day 1, this is not unusual. But by Day 2, the preceding paragraph would imply Persons 1 and 2 would have killed themselves — so they have to had seen a third person. So Person 3 reasons that, having seen no other third person with blue eyes, he is it. By symmetry, Persons 1 and 2 make the same realization. On Day 3, they all kill themselves.

Now this can be iterated indefinitely. For example, in the n=4 situation, on Day 3 Person 4 would expect Persons 1, 2, and 3 to have killed themselves by the above paragraph. But they are still alive, so they each saw another person with blue eyes. Having seen no other set of blue eyes, Person 4 realizes he is it, and by symmetry, Persons 1, 2, and 3 simultaneously make this realization. They all die on Day 4. In general, on Day (n-1), each person with blue eyes expects all the other people with blue eyes to kill themselves, since each person with blue eyes sees (n-1) sets of blue eyes. When nobody dies, every person with blue eyes deduces all the other blue-eyed people see him. As all make this realization on Day (n-1), all die on Day n.

Well, you ask, how about the brown-eyed people? They are irrelevant to this problem. They see n sets of blue eyes, instead of n-1. As such, the decision making of the brown-eyed people is delayed to Day n, when all the blue-eyed people are dead, vindicating the brown-eyed people. The number of brown-eyed people does not change the number of blue-eyed people that are seen by each blue-eyed person, which is what matters.

Another objection that comes up is that the foreigner doesn’t give any new information, because everyone can see who has and has not blue eyes. The key is that there is an exception to this, for the base case n=1, where the foreigner would inform the blue-eyed person that he is it. Without the information, nobody would kill themselves even if there was one blue-eyer on the island. But now, since n=1 happens, n=2 must happen, and so on… In other words, the foreigner provides a logical pathway for someone to find out he has blue eyes. Without the foreigner’s statement, nobody could ever find out that he has blue eyes, thus nobody could ever kill himself, and thus nobody expects anybody to kill himself. If one person could kill himself from blue eyes, then all people with blue eyes are doomed, because everyone with blue eyes would wait until all the other people with blue eyes are dead — until he realizes there is one more set of blue eyes, those belonging to himself.


Tornadogenesis and the Like

Sorry, this is about weather again. I’ll hopefully post juicier, more relatable stuff soon. But this order of business should be taken care of now. Let’s talk about tornadogenesis.

First we have a thunderstorm out there. The ambient air is full of horizontal spin, known as streamwise vorticity or helicity, due to changes in wind speed and direction with height. The thunderstorm updraft tilts the spin vertically, so that the updraft itself obtains a sort of spin. This tornado precursor is known as a mesocyclone. Finally, the thunderstorm downdraft, known as the RFD, rushes downward and mixes with the updraft. The RFD helps to focus the rotation in the updraft. By conservation of angular momentum, when the rotation is focused, it speeds up. Meanwhile, the rising air in the updraft lowers the pressure within the mesocyclone. As rotation speeds up, more air is evacuated through the updraft, further lowering pressures, which increases the pressure gradient between the updraft and the ambient air, increasing rotation and wind speeds, and so on. At a certain point, the pressure lowers enough such that the water vapor in the air condenses, and a characteristic funnel cloud forms. As pressures continue to lower closer to the ground, the funnel cloud lowers to the ground, and a tornado is born.

It is clear from this accepted model of tornadogenesis, that the following are necessary:
1) A thunderstorm with a healthy updraft to tilt spin.
2) Helicity, that is, ambient spin to tilt.
3) A good RFD, to focus rotation in the mesocyclone.

(Note there are other means of tornadogenesis, but we will not explore them here. Most of the strong-violent tornadoes originate in this manner.)

There are three parameters that help us forecast the availability of these three conditions: CAPE, SRH, and LCL heights, respectively.
1) CAPE is a measure of the potential convective energy stored within an air parcel, or its ability to rise spontaneously in the atmosphere. Parcels with higher CAPE are more buoyant. Thus, healthy updrafts typically consist of higher-CAPE parcels.
2) SRH stands for storm-relative helicity, so it is a direct measure of the horizontal spin in the atmosphere.
3) LCL heights measures cloud base heights, and is correlated with the temperature-dewpoint spread at the surface. When cloud bases/LCLs are high, downdrafts take longer to reach the surface, and have more time to cool or dry out on its descent down. These cooler, drier downdraft parcels do not do a good job of interacting with updraft parcels to focus rotation.

From this we see a few points. Helicity determines the initial rotation rate, roughly correlated with v_\theta, the tangential component of tornado wind speeds. CAPE determines the updraft acceleration, correlating with v_z, or vertical wind speeds. Every parcel that exits vertically has to also enter with a radial component, by mass conservation and the no-parcel-penetrates-the-ground condition. Thus an increased v_z entails an increased radially wind speed component v_r. LCLs determines the rotation focus. Increased focus of rotation means increased pressure gradient, or increased v_r. And by the aforementioned mass conservation argument, this means increased v_z.

Recall, again, that in the polar coordinate system
\lVert \mathbf{v} \rVert ^2 = v_r^2 + r^2 v_\theta ^2 + v_z^2 \sim v_z^2 + v_\theta ^2,
the latter arising from the mass conservation argument, and ignoring stupid prefactors. The upshot is, \lVert \mathbf{v} \rVert ^2 is a linear combination of the magnitudes of v_z and v_\theta. Or, put it another way, we can decompose tornado wind speeds into vertical and tangential components, each of which correlates to one or more of CAPE, SRH, or LCL heights. Of course, this is a very, very rough calculation — hardly enough to be taken seriously — but it sheds light on my next sections.

Up until now, we’ve looked at tornadic wind speeds as a whole, which IMO has its limitations because vertical and horizontal wind speeds incur different types of damage. Now from the above paragraph, vertical and tangential (but most horizontal winds are actually tangential in nature near the surface) winds are correlated to CAPE/LCLs and SRH, respectively. So what we can do is look at parameters and see how they relate those to patterns of damage found with horizontal or vertical components of winds. What I’m thinking is that high shear/SRH, low CAPE environments produce different patterns of damage than high CAPE, lower shear environments, given constant LCL. Which has never been accounted for in (E)F ratings, construction practices, or tornado forecasting methods.

High SRH/Low CAPE/Medium LCL
-Structure or garage failure/collapse favored over roof or foundation failure. Roof and foundation bolts are oriented up/down, so horizontal winds are less likely to create problems there. Instead, the winds target the weakest points that face horizontally — the windows and garage. Once tornadic winds get in the structure, they destroy it from within (e.g. wall collapse, generation and penetration of flying debris within the structure). EXAMPLE: 3/2/12 West Liberty, KY EF3; Maps: MLCAPE, 0-1KM SRH, MLLCL
-Structures with weak or nonexistent foundations, for example mobile homes, fare much worse than permanent structures with good foundation. Construction matters a lot here! If key weak points such as the garage are not compromised, damage can be minimized. The “one house destroyed, house next door left untouched” phenomenon can be common. EXAMPLE: 3/2/12 Sawyersville, KY EF3 (Note that though roofing material is removed, the roof frame remains intact. Also, the power poles, which are anchored to the ground, remain standing.)

Moderate-High CAPE/Low SRH/Low LCL
-Vertical motions are great, so roofs can be removed leaving the rest of the frame/structure relatively unscathed. EXAMPLE: 4/3/12 Arlington, TX EF2; Maps: SBCAPE, 0-1KM SRH, MLLCL
-Objects that are relatively light can be picked up and thrown like matchsticks, even if tangential wind speeds aren’t that great. This makes for great footage: 4/3/12 Lancaster, TX EF2

Obviously two regimes with markedly different damage patterns. A couple more worth mentioning.

High CAPE/High SRH/High LCL
Believe it or not, weak tornadoes are often the result. So tornadoes seem to be more sensitive to LCL than they are to either SRH or CAPE. DFW 5/24/11 is a good example. Maps: MLCAPE, 0-1KM SRH, MLLCL

High CAPE/High SRH/Low LCL
This is the violent regime, where EF4-5 damage is often produced. In the other regimes, because damage is so dependent on weak points since the wind is one-dimensional in nature, engineers have many reasons to cap the rating at EF3. Not here. Furthermore, in this regime, the vertical winds can lift a structure off its foundation, while the horizontal winds can destroy the structure – thus the origin of the “sweeping away” phenomenon so characteristic of violent tornadoes. This doesn’t occur in other regimes (instead the structure is “leveled” but not lifted and thrown). This, and the Low-Medium CAPE/High SRH/Low LCL regime, produces the most violent tornadoes. And almost all EF5’s will occur in this regime.