Essence of Genius

“The essence of genius is to know what to overlook”— William James. To solve a problem or explain a phenomenon, analytical genius untangles the relevant from the irrelevant factors. Genius extricates the generative essence to the smallest set of fundamental factors from which the other factors derive. Genius is less dependent on extensive memory (although this aids in the search of possibilities) and more so on the ability to conjure a mental model.

Creativity derives from this ability to extricate the independent factors— those that are not redundant because they don't derive from or to those preexisting, i.e. divergent thinking.


Genius thought process would solve question #22 which appears on the Sigma Society IQ test1 as follows.

22) Several straight line segments are drawn on a plane surface in such a way that their intersecting lines form 1,597 areas that are not further subdivided. What is the minimum number of line segments that must be drawn to form the described pattern?

1The question will probably be replaced since this author has revealed his solution below.

Considering the possible relative orientations of lines, two pairs of parallel lines are required to enclose an area. Whereas, only three non-parallel lines are required to enclose an area.

two pairs of parallel lines    three non-parallel lines

Thus only non-parallel lines shall be in the required minimum solution. Genius has discarded all the possibilities of orientation patterns for this fundamental factor.

The question did not provide a finite extent for the planar surface, thus assuming all line segments are drawn long enough, every line added will intersect every preexisting pair of lines regardless of the orientation chosen. For each preexisting area, the added line either subdivides it or one of its new areas is subdivided by one of the preexisting lines.

extra line subdivides preexisting area    extra line subdivides the new area

Genius has discarded all the possibilities of orientation patterns of non-parallel lines for this fundamental generative rule that each additional line leaves an area (not further subdivided) for each preexisting pair of lines. Thus the total (not further subdivided) areas is the number of pairs that can be chosen from the total number lines (`L`) minus one.

Utilizing high school level math, the number of pairs that can be chosen from `n = L - 1` lines is the combinations where `k = 2`, which can be written in the multiplicative form.

`((n-(2-1))/1)((n -(2-2))/2)` reduced to `(n(n-1))/2`

The answer is `L = 59` (`58.02` rounded up) from the quadratic equation `1597 = (n(n - 1))/2` by employing the quadratic formula.

Abstract Model

Note `(n(n - 1))/2` is equivalent to the triangular number if `n = L - 2`.

In the abstract, each additional line exceeding the initial pair can be conceptualized as adding one unit (e.g. a ● representing a not further subdivided area) of length to each side of a conceptual equilateral triangle three unit equilateral triangle of ● units.

Level of intelligence

The Sigma Society claims that “Men of Noteworthy Talent”, e.g. Rousseau or Lincoln, would have the ability to solve the above problem and all prior problems on the test. Such people would have an IQ of 148— three standard deviations (Sigma III) from the mean IQ of 100, i.e. the 99.87% percentile thus the relative ability of 10,000 ÷ 13 = 769 times greater than the mean.

Visual Pattern Matching IQ

What is this?

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Better Answers to Google's Interview Questions

Apologies if this is quite egotistical, but it is just to demonstrate that most IQ tests are in fact wrong.

Better answers (from me):

Why are manhole covers round?

Man hole covers are round mainly so they can't fall in the hole, but assuming they are quite heavy this also enables them to be rolled. However this could be a disadvantage if risk of theft is a major factor, in which case the ideal shape is an equilateral triangle or the letter 8, or more generally any `N` shapes attached together where the maximum of the minimum grouped dimension (i.e. the "width") is greater than maximum dimension of any of the individual shapes.

You are given 2 eggs...

This is a minimum path optimization question, i.e. finding the shortest distance one needs to travel. The more general answer (if you really want to impress Google), is that if you are given `M` eggs to break for `N` floors, then the answer will always be that maximum drops `= root M N - 1 × M = N` (inv key)(`x^y` key) `M` (`=` key) (`-` key) 1 (`=` key) (`×` key) `M`. For 2 eggs the 2nd root means square root, for 3 eggs cube root. The reason that the `Mth` root of `N` is the shortest path is because it gives you the set with `M` members that are the smallest numbers that can be multiplied together that result in `N`. For example, with 1000 floors and 3 eggs, you get cube root of 1000 is 10, so first you drop in `1000 ÷ 10 = 100` increments, so that is at most 10 drops. Then you drop in increments of 10, then finally increments of 1.

However, note that someone actually deduced a shorter path, and another person derived the generalized equation for 2 eggs and `N` floors, and another alluded to the `M` and `N` generalized form.

How many times a day does a clock's hands overlap?

The answer given was incorrect, because only included the hour and minute hands, and did not account for a military clock. The more general and correct answer that for the minute and hour hands, then answer `= 24 - (24 ÷ n)` where `n` is the number of hours on the clock. For example, for a military clock (13th - 23rd hour on the clock), then the answer `= 24 - (24 ÷ 24) = 23`. Also add the minute and second hands, answer `= 24 × 60 - 1`. Also add the hour and second hands, answer `= 24 × 60 - 1`.

Explain the significance of "bad beef".

The correct answer is that it depends on the context of the field of inquiry. To jump to the conclusion that "bad beef" is the magic hexadecimal file marker (I haven't seen that for 20 years and certainly I don't have the long-term sparse memory recall to have clued in on that), is myopically presumptuous without further qualification. Btw, this is the genre of question that I was marked incorrect on IQ tests, but for which I feel the IQ test examiner was wrong.

You need to check that your friend Bob has your correct phone number...

The answer given was incorrect, and someone else derived the same correct answer that I thought of, which is the phone number is the encryption key, e.g. send the MD5 hash of the phone number, ask Bob to reply whether it matches.

In a country in which people only want boys...

The answer given was incorrect, and someone point out that interestingly a preference for boys will actually lead to more girls in the population! Those with slightly higher probability of reproducing girls, will produce more girls until they produce a boy, but those with slightly higher probability of producing boys, will stop producing boys on their first one. Interesting Biblical rule, that you can not go against God's plan without reaping what you sow. Someone noted the humorous irony.

You are shrunk to the height of a nickel...

I was impressed with someone's answer to the blender question. I was thinking that since our mass was less than a piece of paper, that we had no chance of staying in a set position, nor controlling direction blown, unless we could possibly avoid the air vortex and centrifugal force in the center of the blade (but the spinning would probably kill us), so I was thinking to go to the base of the blade axis and find a rubber gasket to bite and make handles to hold on to. In theory, by the same cube root law, our strength to mass ratio would be orders-of-magnitude increased.

You're the captain of a pirate ship...

The answer given above is incorrect. If you offer all the booty to 51% of the crew, one of them might get motivated to vote against your proposal because they have nothing to lose if your proposal wins (and they might want to try their luck at another proposal that offers them more). Contemplate that deeply! Whereas, the correct answer is to propose that anyone who votes NO will not share in the booty. TADA!

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