By Sanjoy Mahajan
An antidote to mathematical rigor mortis, instructing the way to wager solutions while not having an explanation or an actual calculation
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Additional resources for Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving
Example text
11) Then solve for the coefficients α, β, and γ by reapplying the easy-cases requirements. The b = 0 test along with the h = b/2 easy case, which showed that V = hb2 /3 for an ordinary pyramid, require that α = 1/3. The a = 0 test similarly requires that γ = 1/3. And the a = b test requires that α + β + γ = 1. Therefore β = 1/3 and voilà, V= 1 h(a2 + ab + b2 ). 9)! 9 Integration Use integration to show that V = h(a2 + ab + b2 )/3. 10 Truncated triangular pyramid Instead of a pyramid with a square base, start with a pyramid with an equilateral triangle of side length b as its base.
This piecewise-linear approximation turns the area into a sum of n trapezoids. As n approaches infinity, the area of the trapezoids more and more closely approaches the area under the smooth curve. 2 The table gives the area under the curve in the range x = −10 . . 10, after dividing the curve into n line segments. The areas settle onto a stable value, and it looks familiar. 7, which might arise from 3. 77, which is too large to be 3. Fortunately, π is slightly larger than 3, so the √ area might be converging to π.
3). Easy cases, unlike dimensional analysis, can also eliminate choices like 2/α with correct dimensions. Each tool has its strengths. 7) use the three easy-cases tests to evaluate the following candidates for its value. √ √ √ (a) π/α (b) 1 + ( π − 1)/α (c) 1/α2 + ( π − 1)/α. 2 Plausible, incorrect alternative Is there an alternative to easy-cases tests? 3 Guessing a closed form Use a change of variable to show that ∞ 0 dx =2 1 + x2 1 0 dx . 8) The second integral has a finite integration range, so it is easier than the first integral to evaluate numerically.