Life’s an exercise book,

Messed up in the stationary cupboard.

No hope of tidiness.

I started a company clearing litter from the streets. Business was slow at first, but it’s picking up.

This is a solution to the question I asked on 13 June, what is the rectangle of largest area that can be inscribed in an equilateral triangle of side 1?

Intuitively we might guess that the rectangle concerned is a square but it turns out that it isn't.

Suppose the. base of the rectangle is as shown in Fig. 1. Then we can use trigonometry to show its height is sqrt(3)t and find its area. Fig. 2 shows a graph of the area against t, which is a parabola with a line of symmetry as show, and a vertex P which corresponds to the maximum area. The graph must pass through the point where t = 0, so the width of the rectangle is 1 and its height and therefore area is 0. It must also pass through the point where t = ½ and the width of the rectangle is 0 and therefore its area is again 0. By symmetry the vertex of the parabola must be half-way between these values of t, so the maximum is when t = ¼.

This rectangle has an area which is exactly 50% of the are of the triangle. It's interesting to compare this with the area of the largest square, which is very close, at about 49.7% of the area of the triangle. I drew the diagram below in GeoGebra showing the rectangle in black and the square in brown. You can see how small the difference is.

Suppose we start with an equilateral triangle of area 1 and draw a rectangle in it, how large can the area of the rectangle be?

I went out with an amorous fruit last night. It was a romantic date,

We were doing a crossword at the bar and someone said “Alarm a relative”.

I though what can that mean? I had this vision of frightening aged aunts or something like that. But the answer turned out to be “alpaca”.

For more Mondegreens visit 

I loved you more than worms can say.

9 students and 9 professors attend a dinner at which they sit at a round table. The organisers wish to arrange the seating so that no one is seated between two professors. Is this possible to meet this condition?

Solution: Imagine we have found an arrangement that meets the conditions. Ask every other person to stand up, so 9 people will be standing. Then we cannot have two professors separated by only a single seated person, otherwise that person would be flanked by professors. But this means only 4 of the 9 standing persons can be professors.

However the same reasoning can be applied to the 9 people who are sitting down, so we can only accommodate 8 professors altogether, leaving one professor unaccounted for.

Consequentially no arrangement can exist that meets the given condition.

At the Karma Restaurant we serve just desserts.