# The tallest tower

We have 190 glasses that are used in the game of the "speed stack" which consists of stacking glasses in a pyramidal way as quickly as possible.

The dimensions of the glasses used in the game are 10cm high, 6cm wide at the top and 5cm at the bottom. The glasses have a shape such that if we introduce one inside the other the total height of the set is 11cm since each glass has a stop inside that does not allow them to fit completely.

Two boys are playing to make the tallest tower possible with the glasses. One says that the greatest height is achieved by stacking the pyramid-shaped glasses, as is done in the game since each level of glasses represents 10cm more height and simply with 3 glasses we already achieve a height of 20cm while if we stack the glasses one inside the other, with three glasses we only get a height of 12cm.

The other boy, however, believes he is able to make a taller tower by stacking the glasses inside each other.

Can you tell us who is right and what is the maximum height that can be achieved by stacking the glasses in one of these two ways?

#### Solution

If we stack the pyramidal vessels we get a great height with few glasses but as the pyramid grows we need more and more glasses to go up an additional level. However, if we stack the glasses one inside the other, initially we get low height with the first glasses but the length of the tower increases linearly at the rate of one centimeter per glass.

With the pyramid method we can achieve 19 heights with the 190 glasses: 1 + 2 + 3 + ... + 18 + 19 = 190 which represents a height of 190cm. On the other hand, with the method of the glasses stacked inside each other, we will have a height of 10cm for the first glass plus 1 cm for each additional glass so we will have a height of 10 + 189 = 199cm, so that to achieve the highest height with the number of glasses available, the best tactic is to stack the glasses one inside the other.