House of cards - solution

Tuesday, 28 May, 2019

This is the solution to the monthly maths problem from RTSdécouverte House of Cards

One needs 15 050 cards.

Here is one way to count how many cards are needed. Let the top storey be called Level 1, the next one down Level 2, all the way down to the ground floor which will be Level 100.

Ignoring the ground floor (Level 100) for a moment, each level is made of triangles and each triangle is made up of three cards (we count the horizontal cards as floor, rather than ceiling). Level N consists of N triangles and therefore requires 3N cards.

Thus without the ground floor the number of cards required is

3 + 3 ∙ 2 + 3 ∙ 3 + … + 3 ∙ 99 = 3 ∙ (1 + 2 + 3 + … + 99).

Recalling that the sum of numbers 1 to k is given by the formula k(k+1)/2 this number is easily calculated as:

3 ∙ (1 + 2 + 3 + … + 99) = 3 ∙ (99 ∙ 100) / 2 = 14850.

To this we must add the cards needed for the ground storey which is made up of 100 triangles, but this time each triangle has 2 cards only, making an additional 200 cards. Thus, the total number of cards is

14850 + 200 = 15050. You can read the original problem here: House of Cards