Giza-IQ Test - Solution 4

The volume of a tetrahedron is baseArea * height / 3. If we only extend one of the edges that don’t form the base, as shown in the figure, the maximum height is obtained when the rotating face is perpendicular to the base.

Then, the height coincides with that of an equilateral triangle of side 1: sqrt(3)/2.

The area of the base is sqrt(3)/4. Therefore, the maximum volume of the tetrahedron is:

   Vmax = sqrt(3)/4 * sqrt(3)/2 / 3 = 1/8 = 0.1250.

Just for fun, we can compare this volume with that of a regular tetrahedron. Its height can be calculated as follows:

   h = sqrt( 1² - (2/3)²) = sqrt(5)/3

Therefore, the volume of the regular tetrahedron of unitary edge is

   sqrt(3)/4 * sqrt(5)/3 / 3 = 1/12 * sqrt(5/3) = 0.1076...

Giza-IQ Test - Solution 3

Cut the cone along the line formed by the vertical segment of the string, and open it onto a flat surface.  You obtain a triangle with a rounded base.  The string slips off the cone when the angle at the top of the opened surface of the cone reaches 180 degrees.  That is, when the opened cone looks like a semicircle.

If we take the radius R of the semicircle to be 1, the circumference of the cone base is given by:
  C = 2*pi*R / 2 = pi

The radius of the cone base is then given by
  r = C/(2*pi) = 1/2

The aperture α of the cone is easily calsulated as follows:
  α = 2*arcsin(r/R) = 2*arcsin(1/2) = 60 degrees.

Giza-IQ Test - Questions 9 and 10

Three more geometrical puzzles.  Don’t you like polyhedra?  These are the last two geometrical questions.  Then, I will have for you numerical ones.

1.09    Four regular tetrahedrons are placed inside a regular tetrahedron that has edges of double length.  This is done in such a way that each vertex of the large tetrahedron coincides with one of the vertices of the small tetrahedrons, so that an empty space is left in the middle of the large tetrahedron.  Please describe in words the shape of the empty space.

1.10    Two ants are on the opposite vertices of a regular octahedron.  They choose at random one of the edges and walk on it at the same uniform speed.  Every time each of the ants encounters a vertex, it immediately chooses at random a new edge (that is, it never doubles back onto the edge it has just come from) and walks on it.  What is the probability that the two ants meet after each ant has walked on exactly two edges?

Giza-IQ Test - Solution 2

This is the solution to the second of the two problems I posted on July 11.

Giza-IQ Test - Solution 1

Today, instead of proposing more puzzles, I have decided to begin giving you solutions.

This is the solution to the first of the two problems I posted on July 11.

Giza-IQ Test - Questions 7 and 8

Today’s puzzles are about fitting regular polihedra into boxes.

1.07    A rectangular box can be filled completely with cubes of dimensions 1 x 1 x 1.  When the unit cubes are removed and larger cubes, each having a volume of 2, are placed in the box with their edges parallel to the edges of the box, it turns out that they can only fill 40% of the box. What is a possible set of dimensions for such a box?

1.08    What are the dimensions of the smallest rectangular box that can contain a regular octahedron with unitary edge length?

Giza-IQ Test - Questions 5 and 6

To all French visitors: Happy Bastille Day!  For those who don’t know, on July 14th of 222 years ago, the French Revolution exploded.  That’s why today is the French equivalent of U.S.A’s  July 4th.  I wonder what fraction of such momentous events occurred during summer.  The Russian national day is (was?) October 25th, and the Chinese one is October 1st, but Italy’s is June 2nd, and the Australian one January 26th (which is during the southern summer)...

Anyhow, today I am going to propose two more of my mathematical puzzles.  These two are about Platonic solids: a Tetrahedron and an Icosahedron.  Wikipedia says that many viruses, including the herpes virus, have icosahedral shells. Fascinating  (as an old friend with pointy ears would say)!

1.05    In a regular tetrahedron with unitary edge length, two (and exactly two) edges are increased in length, thereby increasing the volume of the solid.  What is the maximum volume that can be obtained in this way?

1.06    An ant walks on a regular icosahedron with unitary edge length.  It goes from the geometric centre of one face to the geometric centre of the opposite face (the geometric centre of a triangle is the centre of the circumscribed circle).  What is the minimum distance that the ant needs to cover?

Java - Formatting a Sudoku for the Web

I just wrote a small method in Java that somebody might find useful. It converts a Sudoku string into HTML.

Giza-IQ Test - Questions 3 and 4

Here are two more questions to test your intelligence with mathematical and geometrical problems.  The first problem I propose to you today was one of the problems I encountered when in 1967 I attended in Rome the selections for the Italian National Math championship.  I qualified for the national championship and there I reached the fifth place, thanks to which I became part of the Italian team sent to the IX International Mathematics Olympiad.

Giza-IQ Test - Questions 1 and 2

As you might know, in 2010 I joined several High-IQ societies. half a year later, I started working on an Intelligence Test. After developing twenty-three questions, as it often happens to me, I lost interest and left it unfinished. As I don’t think I will ever complete it, I will publish the questions in this blog. To see the correct answers, you will have to send me an email or comment online.

Testing for Rare Medical Conditions

I am not a Medical Doctor, but I know enough about statistics to talk about the validity of [some of] the tests currently used to screen for prostate cancer.

Equinoxes in Canberra

I noticed a discrepancy in Canberra equinoxes that I cannot explain.  Perhaps one of you will be able to shed some light on this puzzle.

Arrays of Functions in Java

A couple of months ago, shortly after publishing the book Sudoku Programming (also see the post in this blog), I asked myself whether I should have written the programs in Java instead of classic C.

I had chosen C because I thought it would be easier for non-programmers to deal with a procedural language rather than with Object Oriented programming, with its class inheritance and operator overloading. But, looking at the number of functions I needed for the C implementation, I began thinking that in Jva the implementation would probably have been simpler. I simply had to re-develop Solver and Generator in Java. I simply had to do it.

One of the first problems I encountered was how to implement in Java an array of function pointers.

Sudoku Explained

I have just released the book “Sudoku Explained”.

Casualties

Another Australian soldier has died in Afghanistan.  It saddens me to think that a hail of nails and ball bearings on a dusty road killed a young, strong, and courageous man.  Like most people, I know what it means to lose somebody you care about, and my thoughts go to Sergeant Wood’s wife.

Refugees

The members of Mensa Australia exchange ideas via a mailing list.

On May 1^st, one of the members posted the following message, under the heading “cost-benefit of refugees”:

To an Economist, cost and benefit is everything.