# Browse Classroom Capsules and Notes

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A standard square in the xy-plane is a unit square whose corners have integer coordinates.  This note shows that when a polyomino consisting of an odd number of standard squares is cut into...

This note shows that graphs with $$n$$ vertices containing no complete graph with $$r$$ vertices, have no more than $$(r - 2)n^2/(2r - 2)$$ edges , for $$r \geq 2$$.

Let $$X$$ be a nonempty metric space without isolated points.  If $$G$$ is a countable intersection of open sets, the author shows that there is a function $$\phi (x)$$ that is continuous...

Using a result on periodic continued fractions, the author presents a rational function method of approximating square roots that is faster than Newton's method.

Let $$G$$ be a group and $$C(a)$$ be the centralizer of $$a\in G$$.  The author studies the properties of the skew centralizer $$B(a)=\{x\in G : xa=a^{-1}x \}$$ and the reversing...

A real matrix is called square-palindromic if, for every base $$b$$, the sum of the squares of its rows, columns, and four sets of diagonals (as described in the article) are unchanged when...

The author clarifies the wording of Cramer's rule, sidestepping a common misconception.  The Kronecker-Capelli theorem is introduced to help see Cramer's rule in a more complete...

Like Pascal’s triangle, Faulhaber’s triangle is easy to draw: all you need is a little recursion.  The author demonstrates that the rows are the coefficients of polynomials...

The author considers the effects of inflation and interest rates on computing the present value of \$100 three years from now.

It is well known that maximal ideals in the ring of continuous functions on the closed interval $$[0, 1]$$ are not finitely generated. Less well known is the fact that these ideals are not...