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Which Is The Quickie and Which Is The Trickie?

In the 1950's, Mathematics Magazine regularly published two problems sections, called "Quikies" and "Trickies" respectively. By their definition, a "Quickies" were problems that seemed solvable by "laborious methods, but which with the proper insight may be disposed of with dispatch." In contrast, "Trickies" were problems whose "solution depends upon the perception of the key word, phrase or idea rather than upon a mathematical routine." Consider these two examples...

[Q1] Prove that (21/20)100 > 100. (No tables or calculators allowed.)

[Q2] Find a function of a variable such that if the variable varies in a Geometric Progression the function varies in an Arithmetic Progression.

And which do you think is the Quickie? The Trickie?


Source: Adapted from Mathematics Magazine, 1954, pp. 37-38 and 1957, p. 173.

Hint: Rather than provide a hint, I will tell you the first is the Quickie and the second is the Trickie...does that help, given their definitions?


Solution Commentary: The suggested solutions (as you may find others possible) are:

[S1] First show (21/20)50 > 10? Write (21/20)50 = (1+1/20)50. Expand latter using Binomial Theorem to get a sum of positive terms: 1 + 2.5 + 3 + 2.4 + 1.4 +.... > 10.

[S2] Think about the properties of the logarithmic and exponential functions. For example, let y = ax. Then as x varies as an arithmetic progression (m, m+c, m+2c,...), what happens to y?