Puzzles

A few examples are:

I have in mind a number which when you remove the units digit and place it at the front, gives the same result as multiplying the original number by 3. Am I telling the truth?

When halfway around the corner, the grand piano will look like this:

The problem then is finding the largest area rectangle which fits in the highlighted triangle: an isosceles triangle where the base is twice the height. Let the base be 2 and the height be 1.

If the height of the rectangle is \(h\), then its width is \(2(1-h)\). Therefore its area is \(2h-2h^2\). By differentiation, it can be seen that this is maximum when \(h=\frac{1}{2}\), which means that ratio of the rectangle's length to its width is 2:1.

If the corner was not a 90° angle, then what is the largest area rectangle which could fit round it?

On hearing the eighth whistle, the dog will be running at 512m/s, faster than the speed of sound. Hence, no more of the master's whistles will reach the dog. However, the dog will catch up with the seven whistles she has already heard and hear them again. So fifteen whistles will be heard in total.

How much time will the dog have been running for when she hears the final whistle?

If you start with \(n\) cups with one pea in each, how many moves is it possible to make?

One of the terms in the series will be \((x-x)\), so the product will be zero.

A 1, 3, 4, 6 & 6

B 1, 1, 1, 3 & 9

C 1, 2, 2, 4 & 6 or 2, 2, 2, 2 & 7

A and B have only one solution. C has two solutions. For which other numbers a, b, c and d does the problem

"Find five one-digit positive integers which have a mean of a, mode of b, median of c and a range of d."

have more than one solution

Name the regions as follows:

\(E\) is a 1×1 square. Placed together, \(A\), \(C\), \(G\) and \(I\) also make a 1×1 square. \(B\) is equal to \(H\) and \(D\) is equal to \(F\).

Therefore \(B+E+F=A+C+D+G+H+I\). Therefore the hatched region is \(C\) larger than the shaded region. The area of \(C\) (and therefore the difference) is \(\frac{1}{4}\).

What is the difference between the shaded and the hatched regions in this dodecagon?

As our shape is a triangle, the 4cm and 5cm sides must be adjacent. Call the angle between them be \(\theta\).

The area of the triangle is \(\frac{1}{2}\times 4\times 5 \times \sin{\theta}\) or \(10\sin{\theta}\). This has a maximum value when \(\theta=90^\circ\), so the largest triangle has and area of 10cm² and looks like:

What is the largest area triangle with a perimeter of 12cm?