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Dick's Math Puzzle PageA collection of mathematical puzzles, some original, with fully worked solutions.
http://www.mod2.org/
A collection of mathematical puzzles, some original, with fully worked solutions.
http://www.mod2.org/
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Dick's Math Puzzle Page | mod2.org Reviews
https://mod2.org
A collection of mathematical puzzles, some original, with fully worked solutions.
mod2.org
Largest and Smallest Conics
http://www.mod2.org/ellipse.html
Solution to "Largest and Smallest Conics". This puzzle is solved using coordinate geometry and invoking the properties of the AFFINE transformation. X = a*x b*y c. Y = d*x e*y f. This transformation has the property that the ratio of any two areas is unchanged by transformation. Furthermore:. Straight lines transform to straight lines. Ellipses (and circles) transform to ellipses. Consider an equilateral triangle of side 1 unit. The area of the inscribed circle is Pi/12. Back to Dick's Math Puzzle Page.
The Village Surveyors
http://www.mod2.org/sur_sol.html
Solution to "The Village Surveyors". The solution to this problem depends on the properties of the affine transformation. This transformation has the form. X = a*x b*y c. Y = d*x e*y f. It transforms straight lines into straight lines and parallel lines into parallel lines and one of its properties is that the ratios of areas of figures remains the same after transformation. It has 6 parameters, which means that any triangle can be transformed into any other triangle. We now have to place the villages.
Sets of Points separated by Rational Distances
http://www.mod2.org/rat_pnts.html
Solution to "Sets of Points separated by Rational Distances". J Leech's solution is the nine points whose coordinates are:-. 0,0),(182,0),(209,0),(391,0). 120,0),(120,182),(120,209),(120,391),(60,195.5). These points can obviously be rotated and rescaled without making any of the distances irrational. If this is applied to a triangle whose sides are rational then cos(theta) must be rational. If you divide both sides by b. Thus repeated applications of this process produce a family of related sets.
A Long Street
http://www.mod2.org/str_sol.html
Solution to "A Long Street". A solution for mere mortals would run something like this:-. Let N be the total number of houses in the street. Let n be the number of the friend's house. Using the standard formula for the sum of an arithmetic progression. The sum of the houses below n is n*(n-1)/2. The sum of the houses above n is (N-n)*(N n 1)/2. Equating and simplifying gives. Replace N by (p-1)/2 and n by q/2 gives. We use a=c=3, b=d=2 to give a second solution, p = 17, q = 12 and.
Looking Glass Zoo
http://www.mod2.org/zoo_sol.html
Solution to "Looking Glass Zoo". There were three nephews and two nieces. A boy won. He got two pairs right but interchanged two sexes. Satisfactory proofs of this result are hard to find. The author gave a proof using matrix theory, but even he omitted the crucial point, just saying 'It has been proved that'! I used a "brute force" computer program as follows:-. Not at all politically correct). Back to Dick's Math Puzzle Page.
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Dick's Math Puzzle Page
Dick's Math Puzzle Page. This page contains the best puzzles that I have found over many years of searching plus a few original ideas of my own. I offer them in the hope of being offered similar puzzles in return. The puzzles are not all original although I have extended some of them and some of the solutions are mine. My address is:. Dividing the Flower Bed. This is an original puzzle. Let sq(x) imply the positive square root of a positive number. Consider a sequence of functions as follows:-. Slightly ...
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