Two-valued number pyramids

Number 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...

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Bibliographic Details
Published inMathematische Semesterberichte Vol. 69; no. 1; pp. 1 - 18
Main Author Kastenholz, Helmut
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2022
Springer Nature B.V
Subjects
Binomial coefficients
Mathematical analysis
Mathematics
Mathematics and Statistics
Mathematik in Forschung und Anwendung – Mathematical Research and Applications
Pyramids
Colored tilings
97H20
Number pyramids
52C20
Discrete calculus
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ISSN0720-728X
1432-1815
1432-1815
DOI10.1007/s00591-021-00307-6

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Summary:Number 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 step number pyramid can have at most 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.
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ISSN:0720-728X
1432-1815
1432-1815
DOI:10.1007/s00591-021-00307-6