Download Street-Fighting Mathematics: The Art of Educated Guessing by Sanjoy Mahajan PDF

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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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.

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