Elementary Geometry

Using only elementary geometry, find the measure of angle x.



This problem is harder than it looks.

New Math

Here is some "New Math" by Craig Damrauer:











For more "New Math" go here.

Archimedes' Cattle Problem

This problem, which Archimedes sent to the mathematicians of Alexandria to investigate, involves computing the number of cattle in a herd of the sun god from a given set of restrictions. It was originally stated in the form of a 44 line poem in Greek. The following is an English translation:

"If thou art diligent and wise, O stranger, compute the number of cattle of the Sun, who once upon a time grazed on the fields of the Thrinacian isle of Sicily, divided into four herds of different colours, one milk white, another a glossy black, a third yellow and the last dappled. In each herd were bulls, mighty in number according to these proportions: Understand, stranger, that the white bulls were equal to a half and a third of the black together with the whole of the yellow, while the black were equal to the fourth part of the dappled and a fifth, together with, once more, the whole of the yellow. Observe further that the remaining bulls, the dappled, were equal to a sixth part of the white and a seventh, together with all of the yellow. These were the proportions of the cows: The white were precisely equal to the third part and a fourth of the whole herd of the black; while the black were equal to the fourth part once more of the dappled and with it a fifth part, when all, including the bulls, went to pasture together. Now the dappled in four parts were equal in number to a fifth part and a sixth of the yellow herd. Finally the yellow were in number equal to a sixth part and a seventh of the white herd. If thou canst accurately tell, O stranger, the number of cattle of the Sun, giving separately the number of well-fed bulls and again the number of females according to each colour, thou wouldst not be called unskilled or ignorant of numbers, but not yet shalt thou be numbered among the wise.

But come, understand also all these conditions regarding the cattle of the Sun. When the white bulls mingled their number with the black, they stood firm, equal in depth and breadth, and the plains of Thrinacia, stretching far in all ways, were filled with their multitude. Again, when the yellow and the dappled bulls were gathered into one herd they stood in such a manner that their number, beginning from one, grew slowly greater till it completed a triangular figure, there being no bulls of other colours in their midst nor none of them lacking. If thou art able, O stranger, to find out all these things and gather them together in your mind, giving all the relations, thou shalt depart crowned with glory and knowing that thou hast been adjudged perfect in this species of wisdom."

Can you solve the first part and thereby "not be called unskilled or ignorant of numbers"?

Can you solve the second part and thereby "depart crowned with glory and knowing that thou hast been adjudged perfect in this species of wisdom"?

[Note: Part 1 is definitely doable. Part 2 is considerably more difficult.]

A Father, Mother, and Son

I have been in a "puzzle mood" lately. Here is one more. This one is a bit more difficult than the last two.

The ages of a father, mother, and son add up to 70. The father is 6 times as old as the son. When the father is twice as old as the son, the combined age of the three is 140. How old is the mother?

Multiplication and Addition

We all know that 2 + 2 = 4 and 2 x 2 = 4. We also know that 0 + 0 = 0 and 0 x 0 = 0. Are there any other positive numbers A and B such that A + B = A x B = C?

[As before, please do not post an answer as a comment. Feel free to email me a response. To prevent redundancy, this applies to all puzzles and problems hereafter.

~Armando]

Pythagoras's Student

"One story claims that a young student by the name of Hippasus was idly toying with the number √2, attempting to find the equivalent fraction. Eventually he came to realize that no such fraction existed, i.e. that √2 is an irrational number. Hippasus must have been overjoyed by his discovery, but his master was not. Pythagoras had defined the universe in terms of rational numbers, and the existence of irrational numbers brought his ideal into question. The consequence of Hippasus’ insight should have been a period of discussion and contemplation during which Pythagoras ought to have come to terms with this new source of numbers. However, Pythagoras was unwilling to accept that he was wrong, but at the same time he was unable to destroy Hippasus’ argument by the power of logic. To his eternal shame he sentenced Hippasus to death by drowning."

~ Simon Singh, from Fermat's Last Theorem (1998)

Tupper's Self-Referential Formula

Jeff Tupper "discovered" the following remarkable formula that when plotted under certain conditions produces the graph shown after. Go here for more information.