Question

You want to build a stair-like structure out of toothpicks. Is it possible to use a total of 2628 identical toothpicks to create a stair-like structure?

Answer 1

For first step of the stair, we need 1 toothpick.

For second step of the stair, we need 2 toothpicks.

For third step of the stair, we need 3 toothpicks.

Proceeding in this way, for n-th step of the stair, we need n toothpicks.

Let we can make x steps of the stair using 2628 identical toothpicks.

For x steps of the stair, we need (1+2+3+4+5+.....+x) toothpicks.

Then by the given condition we have,

1+2+3+4+5+....+x = 2628

i.e., x(x+1)/2 = 2628

i.e., x(x+1) = 2*2628

i.e., x²+x = 5256

i.e., x²+x-5256 = 0

i.e., (x+73)(x-72) = 0

i.e., x = -73, 72

Since the number of steps can't be negative, so x -73.

Therefore, x = 72

Hence, it is possible to build a stair-like structure using a total of 2628 identical toothpicks and the number of steps of that stair is 72.