Two-valued number pyramids
AbstractNumber pyramids are common in elementary school mathematics. Trying to express the value of the top block in terms of the values at the base leads to the binomial coefficients. It also seems natural to ask for the maximal number of odd numbers in a number pyramid of a given size. The answer is easy to state, but the proof is nontrivial: A $$k$$ k step number pyramid can have at most $$\left\lfloor\frac{k(k+1)+1}{3}\right\rfloor$$ k ( k + 1 ) + 1 3 odd numbers, which equals two thirds of the number of blocks rounded to the nearest integer. All maximal and almost maximal solutions are given explicitly. To this end, we rephrase the question in terms of colored tilings. In the outlook we present relations to other—mostly geometric—subjects and problems.
