DejMar

It very well may be possible that a right triangle with an hypotenuse of sqrt(24) inches to have an area of 13 squared inches, yet not a Euclidean right triangle. One might consider verifying the assertion as it relates to right triangles that are hyperbolic.

There are more solutions than the one given by thalia: Other solutions may (and I believe) exist.

Buddyboy3000, However, as of yet, I, too, have not found the arrangement of the numbers in the four by four grid.

I would declare you correct that I made an err. It is likely I transcribed the incorrect results from working of the problem.

One possible solution:

"It goes to the center every time ." -- BMAD What is not made perfectly clear is whether the center is that of the original unfolded paper, or that of the folded paper. For each fold, the center of the external shape will be located at a different point.

It appears the shortest overall length of spinneret thread would be ≈ 6.196152 meters. The red lines in Plainglazed figure would each be √3/3 meters in length, the green lines would each be √3/6 meters in length, and the central yellow line would be (1 - √3/3) meters in length. This should total (3 + 9√3)/3 meters.

Euler? Euler? Euler?

Just a request for clarification or correction to the following assumptions and implications. An assumption is that the ten unrelated people at the beginning of the problem are five males and five females. An assumption is that the resultant birthling, if any, enters society immediately the day following the mating, with equal chances of being involved in mating as those already existing in the society. And, and an assumption is made that conception occurs automatically where the two parents are at least 10% related. An implication is that each adult only performs the ritual of mating no more than once per day, and due to there being a possibility a gender surplus or shortage respective of the other gender, not every adult will mate on a given day. An implication is that incestous matings can occur, though the result of that mating is no offspring.

tojo928's proposal is better than the simple design of connecting the nine towns with orthogonal roadways. An improvement to his solution is

It can be argued that 2, 3, 5, and 7 are members of the sequence. Where the decimal expansion of n is 1, k = 1, and thus the number remains simply the prime. The integer sequence A097227 should, with the exclusion of these terms, state that the there are a minimum of two digits in each term, i.e., k > 1. Though the OEIS tries to be thoroughly correct, it is not always.

What is the divisibility rule?

I would rather not attempt to solve this problem with the lack of information and ambiguity in the problem.

There are more than 24 arrangements of the operands and operators ...