### geometry.puzzles - brief descriptions

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#### 1996

1. Bicycle Wheel () 11/18/96
Name the path which is traced by a point on the wheel of a bicycle as the bicycle moves in a straight line on a leve surface. Where can I find a sketch of it?

2. Congruent Triangles (Wes McCollom) 11/08/96
My geometry book says that, for two triangles to be congruent, you must prove that the hypotenuse AND one leg are congruent. I disagree. It seems to me that you only need prove that the hypotenuses of the two triangles are congruent. [Pythagorean Triples]

3. Cable around the Equator () 11/04/96
Ma Bell wants to place a telephone cable around the equator. She adds 50 feet to the length of the cable beyond what is required. This slack in the cable allows the cable to be strung up above the ground. How high up from the surface of the earth will the cable stand?

4. Concave Triangle Formula (John Erickson) 11/01/96
If you place three circles of the same diameter so that each circle is touching the other two circles and all three circles together form a "triangle," there is an area in the center, outside the circles, in the form of a "triangle" with concave sides. What's the formula for determining the area of this figure?

5. Logic Puzzles (Lesley Breen) 10/17/96
I need quick and simple - but tricky - questions for a grade 8 class. I have only 15 minutes within which to let the students solve it with my help.

6. Brain Teasers (Lesley Breen) 09/24/96
Know any tricky brain teasers (puzzles) for high school math students?

7. Announcing OOG, The Object Orientation Game v1.1 (Joe Vitale) 07/10/96
OOG, The Object Orientation Game, by MCM Productions is the first to present 4 different tiling puzzle games in a single shareware application. OOG challenges you to try and solve the tiling puzzles of Tangrams, Pentominoes, Hexagons and Polyominoes.

8. Hughmoar County Puzzle (Tyler Stevens) 06/11/96
In Hughmoar County, residents shall be allowed to build a straight road between two homes as long as the new road is not perpendicular to any existing county road. Story problem ensues....

9. Probability of a Triangle (Pat Ballew ) 05/29/96
a) If a unit length segment is randomly broken at two points along its length, what is the probability that the three pieces created in this fashion will form a triangle? b) If the length is broken at a random point, and then one of the two pieces is randomly selected and broken at a random point on its length what is the probability that the three pices will form a triangle?

10. Perimeter of Triangle (Reggie Nelson) 05/22/96
If the perimeter of a right triangle is 10 units, and one of the legs times the hypotenuse equals the other leg squared, what are the measurements of the 3 sides?

11. Circles (Mike Slack) 05/20/96
How many sides does a circle have?

12. Cut a Rug (SATS5) 05/12/96
You have two rugs, a 10X10 and a 1X8. Your task is to create a 12X9 rug. You may cut only one of the rugs, such that there are three remnants remaining after the cut is made.

13. Triangulations of Polyhedron (Louxin Zhang) 04/02/96
In a plane, any triangulation of an n-node convex polygon has n-2 triangles. However, in 3-dimensional space, different tetrahedrizations of an n-node convex polyhedra may have different numbers of tetrahedrons. I am interested in results relating the lower bound of the number of tetrahedrons in any tetrahedrization of an n-node convex polyhedra P to degrees of nodes in P.

14. Right Tetrahedron with Integer Edge Lengths? (Daniel A. Asimov) 01/03/96
Let a, b, c be positive real numbers. Can they be chosen so that all six distances among the points (0,0,0), (a,0,0), (0,b,0), (0,0,c) are integers? OR, boiled down to its essence: Do there exist positive integers a, b, c such that the three quantities a^2 + b^2, b^2 + c^2, and c^2 + a^2 are all perfect squares? [Klee and Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory; Dickson, Theory of Numbers]

15. Equalizing Apparent Angular Separation of Pairs of Points (Stephen Buckle) 01/03/96
Given two pairs of points in 3D Euclidean space - call these P1 and P2, and P3 and P4, where each Pi is described by a column vector of Cartesian coordinates [ xi yi zi ]^T - find a fifth point Q from which the apparent angular separation between P1 and P2 is equal, or most equal, to the angular separation between P3 and P4.

#### 1995

1. Four legged stool problem (Jeff Wisnia) 12/22/95
When a 4 legged bar stool (with equal length legs) sits on an uneven floor it usually wobbles. However, I've noted that rotating the stool around its axis of symmetry will always produce a situation where all four legs touch the floor at the same time. Naturally, this location is found in less than 90 degrees of rotation. I had a math whiz tell me once that this will always occur if one imagines the legs terminate in sharp points and as long as the floor has no discontinuities or extremely exaggerated peaks in it. Why must this occur?

2. Triangle construction (Jose Dominguez) 12/05/95
Given three segments of known length: a, b and c, construct a triangle whose bisectors are the given segments.

3. Rectangle Construction (Maky Manchola) 11/28/95
Construct a rectangle given its perimeter and the length of one of its diagonals.

4. Prove:PE+PD=AG (Steve Heintz) 11/19/95
GIVEN: Isosceles triangle ABC, segment AG is perpendicular to segment BC, segment PE is perpendicular to segment BC, and segment PD is perpendicular to segment AB. PROVE: PE+PD=AG

5. Square with Area A (Maky Manchola) 11/14/95
Given a square with area nA (n is a positive integer), construct a square with area A.

6. Triangle to Square (David M. MacMillan) 11/07/95
Given an equilateral triangle, how can you cut that triangle so that the resultant pieces may be reassembled into a square? [Martin Gardner - The Scientific American Book of Mathematical Puzzles and Diversions / "536 Puzzles and Curious Problems" by Dudeney (Scribners, 1967)]

7. Drawing a tangent to a circle (Richard William Weyhrauch) 11/05/95
Problem: Construct the tangent to a circle at a given point using only a straightedge. After you describe the construction please give a proof.

8. Medians of a Triangle (Maky Manchola) 11/03/95
Given the lengths of the medians of a triangle, construct the triangle.

9. Prove:PE+PM=AN (Steve Heintz) 10/29/95
Given: Rectangle ABCD with P any point on line AB. line PE is perpendicular to line BD, line PM is perpendicular to line AC, line AN is perpendicular to line BD. Prove: PE+PM=AN.

10. Largest Equilateral Triangle Inside of Square Torus (Daniel A. Asimov) 10/19/95
Consider the "square torus" T constructed from a unit square [0,1] x [0,1] by identifying (x,0) with (x,1) for each x in [0,1] and identifying (0,y) with (1,y) for each y in [0,1]. What is the smallest equilateral triangle that's too large to be a subset of T (without intersecting itself)? Or: What is the side of the smallest equilateral triangle which, no matter how it is placed in the plane, must contain (in its interior or on its boundary) at least one point of the form (K,L) where K and L are integers ?

11. Sum of angles of a triangle (Tom Chu ) 10/19/95
Let ABC be a triangle and P, Q, R are points on AB, BC, and CA as shown. Start on A and "walk" towards B. At B turn towards C. You have turned an angle equals to PBC which is (180 degree - angle ABC). Proceed or walk towards C, and at C turns towards A. You have turned an angle equal to QCA which is (180 degree - angle BCA). Proceed to A, and at A and turn towards B. You have turned an angle equal to CAP which is (180 degree - angle CAB). In the whole process, you have turned three times and made a complete circle of 360 degrees. Therefore 360 = (180 - angle ABC) + (180 - angle BCA) + (180 - angle CAB) or angle ABC + angle BCA + angle CAB = 180 degrees. Does the above constitute a proof?

12. Paper folding a 30-60-90 triangle (Pat Ballew) 10/17/95
Just looking at an Origami book and saw a really quick method of folding a 30-60-90 triangle from a square or rectangular piece of paper. It made me wonder: a) what other angles are possible with simple paper folds; b) are there any that are impossible?; c) how many different ways can you fold a 60 degree angle? [patty paper folding]

13. Rubik's Cube solution (Peter Ejemyr) 10/16/95
Is there anyone who can point me to an easy-to-understand text about how to solve Rubik's Cube (3x3x3). Or anyone got the pamphlet with solution that use to come with the cube? [two solutions]

14. Ellipse inscribed in a circle (Adriaan Canter) 10/13/95
In the unit circle centered on the origin, I plot any two points. It appears to me that I can draw an ellipse such that the ellipse intersects these two points and is tangent to the circle at two points; ie it is inscribed by the circle. Note that the major and minor axes of the ellipse are inclined with respect to the x and y axese and pass through the origin. It seems I can always finagle a sketch which shows this, but I can't figure out how to prove it, or how to formally construct the ellipse. I know the equation for an inclined ellipse centered on the origin is Ax^2 + Bxy + Cy^2 = 1. Two points isn't enough to solve for three constants, and I'm at a loss to figure out any other constraints. Can anybody see how to do this, or prove I'm wrong?

15. Prismoid rule (Vinko Vrsalovic) 10/11/95
How can you apply the "Prismathoid rule" (I'm not sure if that's the way to say it in English) to a cardioid ? [Prismoid rule]

16. Standard Math Symbols in ASCII (Gerald D. Brown) 10/10/95
Does anybody know of a list of recognized ASCII character strings used to denote standard math symbols?

17. Foundation problem (Maky Manchola) 10/07/95
There is a theorem which states that any mathematical system is incomplete, roughly meaning that given a set M of undefined terms and axioms, there exists a statement which can be neither proven nor disproven using statements from (or derived from) M. Provide an example that depicts incompleteness of some geometry. Justify. [There is no such theorem as quoted below. It is missing a key ingredient, namely that the system must be sufficiently rich; i.e., it must contain within it a set equivalent to the integers. This, of course, is the well-known Godel Incompleteness Theorem.]

18. Construction problems (Maky Manchola) 10/04/95
(1) Construct triangle ABC given angle A, side a , and a segment b+c equal in length to the sum of the triangle's other two sides. [two possible solutions]. (2) Construct an angle whose measure is 3 degrees.

19. Trisecting (or more) an angle (Richard Pennock Jr.) 09/29/95
Is there a problem in Euclidean geometry dealing with trisecting an angle using only a straight edge and a compass? I seem to recall that it can not be done. I think that I have developed a method which will allow me to subdivide an arbitrary angle into any number of equal angles (including three) provided the starting angle is not equal to 180 degrees. (It may work for 180 degrees, I just haven't tried 180 degrees yet.) If all of the above is true, what do I do now?

20. Object related to a triangle (Pat Ballew) 09/25/95
In the Law of Sines, the relation is usually given as a/Sin A = b/ Sin B = c/ Sin C but these ratios also constitute the measure of a geometric object related to the triangle. What is the object ?

21. OOG, The Object Orientation Game (Joe Vitale) 09/19/95
MCM Productions announces OOG, The Object Orientation Game - an MS Windows application for playing & solving various polyform puzzles.

22. Projecting Ellipsoids: Do You Get Ellipses? (steve gray) 08/05/95
To all those who have thought about the question of viewing an ellipsoid : The question was whether a general perspective projection of an ellipsoid is always an ellipse. In other words, given a fixed ellipsoid (with three arbitrary axes) in space and a fixed external point, draw a bundle of lines from the point tangent to the ellipsoid; intersect this cone of lines with a plane. Is this curve of intersection an ellipse? [all ellipsoids are affinely equivalent to spheres, and the projection of a sphere is always a circle]

23. A problem in coor geometry (Mark Tong) 08/02/95
Let Q be the graph of y = f(x), where f is differentiable. Suppose that Q does not pass through the origin O. Let J be the point on Q closest to O. Prove: the ray OJ is perpendicular to Q at J.

24. Point Inside Rectangle - distance from vertex (Archie Benton) 07/20/95
A point is selected inside a rectangle such that its distance from one vertex is 11 cm, its distance from the opposite vertex is 12 cm, and its distance from a third vertex is 3 cm. Its distance, in cm, from the fourth vertex is: a) 20 b) 16 c) 18 d) 14 e) 13 "

25. Projective Geometry Question (S. G.) 07/13/95
I have the following problem to solve: We have an observation point p and plane P. The minimum distance between p & P is given : d. A rectangle with sides a&b (given) has the orientation in 3D such that one of its corners is located at point O on P which is perpendicular to p (i.e d=|p-O|) . The line drawn from p to the intersection of the diagonals of the rectangle is perpendicular to the rectangle. Q: The rectangle is projected onto the plane P from point p . What is the angle between the sides of the projection of rectangle which intersect at O ?

26. Elephant Puzzle (Bob Hesse) 07/05/95
The problem in this elephant puzzle is to fold the triangles into a tetrahedron so that the head, body, and tail of the elephant match up, and so that the remaining face is blank. The blank face can either be the actual blank triangle, or it can be the opposite face of one of the elephant-part triangles.

27. Circle Reflection (Steve Gray) 06/29/95
Given a circle and two points A and B exterior to it, find that point on the circle C so that AC and BC make equal angles with the tangent to the circle passing through point C.

28. Volume of a Geodesic Dome (Jason Acker ) 06/28/95
Can any one describe for me how one would determine the volume of a geodesic dome as a function of the number of sides it has? I've made the assumption that all the sides are triangular with 60 degree angles. How much more difficult would it be to use polygons and triangles instead of all triangles?

29. Looking for Info on G. Spencer-Brown (tbev) 06/25/95
Does anyone know where I can find more information on the mathematical work of G. Spencer-Brown? He wrote a book entitled THE LAWS OF FORM that dealt with the fundamental arithmetic underlying boolean algebra. [discussion of Spencer-Brown's mathematics, generation]

30. Larger Infinity (Doug D. Nichols) 06/19/95
A line segment is made up of an infinite number of points because no matter which two points you pick, as long as the two points are not the same point, you will be able to place a point exactly half way between them. A lint also is made up of an infinite number of points since it continues for ever and ever in both directions. My question is this....1) If they both contain the same number of points (an infinite amount) why are they not considered to be the same length? ...2) If they are not the same length then should we be able to argue that there is an infinity larger than infinity...or is that possible, to have one infinity larger than the other? [even two points can be any distance from each other, even though they are always two points, so number of points is one thing, and length or distance is something else / discussion of lines, points, infinities]

31. Volume question (Jason Acker) 06/01/95
I need to calculate the volume of an N-sided (N-vertices) object inscribed in a sphere. Does anyone know of a formula which will calculate this? [polyhedra]

32. Test for 'Euclideanitiy' (Juan Miguel Vilar) 06/01/95
We have a finite set of points {a1, a2,...,an}, and the distances between them. We want to know if there is an Euclidean space in which there exists other n points {p1, p2,....,pn}, such that d(ai,aj)=d(pi,pj) - whether there is an algorithm to test whether they can be embedded in an Euclidean space. We don't need to specify the dimension or the coordinates, etc., I only need to know whether such space exists. ["Cayley Menger Determinants" - K. Menger or T. Havel: "Some examples of the use of distances as coordinates for Euclidean Geometry" or related works on 'Distance Geometry' / Conway gives algorithm]

33. Shortest Crease (BPManning) 05/24/95
The Shortest Crease: you have a sheet of paper that is 6 units wide and 25 units long, placed so that the short side is facing you. Fold the lower right corner to touch the left side. Your task is to fold the paper in such a way that the length of the crease is minimized. What is the length of the crease? In-creasing questions about creases: 1) what is the envelope of the crease? 2) how long does the long side have to be for the shortest crease to be the one obtained by folding onto the opposite corner? 3) what if you are allowed to fold anywhere on the line containing the long side? 4) what is the shortest crease when you fold a vertex of a triangle onto a side?

34. Quarter circles in a square (Steve Miller) 05/12/95
There is a square with sides measuring 1 unit. Each of the four corners of the square is the center of circles whose radii are 1 unit long. A diamond-shaped figure located in the center of the square is created by the intersection of all four circles. What is the area of the diamond-shaped figure?

35. Pyramid Problem (Aaron Koller) 04/21/95
I need to build a pyramid with a rectangular 3' by 6' base and a volume of 6 cubic feet. The line that contains the apex of the pyramid is perpendicular to the plane of the base. Therefore, the altitude of the pyramid is exactly 1 foot. I need to know the angles formed at the intersection of two NON-BASE faces (there should be 2 different values for a rectangular base) so I can bevel the faces correctly.

36. Chess Board Puzzle (Evelyn Sander) 04/19/95
You are given a chess board and a some dominoes, each of which covers exactly two squares on the chess board. You can cover the board with exactly 32 dominoes. If you remove the two opposite corners of the board, can you cover the remaining squares with 31 dominoes? [variation]

37. Piet Hein's Super Ellipse (Paul Flint) 04/19/95
Request for information and references on the subject of the "super-ellipse" or rectangular oval shape described by Piet Hein [of SOMA fame] some years ago. [equation like an ellipsoid / Martin Gardner's Mathematical Carnival and other books / Scientific American]

38. Ladder puzzle (Evelyn Sander) 04/18/95
A person is standing 8 feet up on a 12 foot ladder. The ladder slides down the wall. What is the path that the person follows while falling? [an ellipse in general and a circle when the standpoint is the mid-point of the ladder]

39. Origami - request for sources (Dale Henry) 04/10/95
Can anyone could help me in a geometry project I'm doing at school? It deals with applying geometry in the "real world" and demonstrating geometric concepts. I decided to do origami because I know it is a geometric art that has many concepts in it. Is there anyone out there that can help me find some geometric concepts in origami? [book suggestions]

40. Puzzles & teaching (Levy4) 03/24/95
Does anyone have any interesting puzzles that would be good for teaching math concepts to middle school and high school students? [Discovering Geometry from Key Curriculum Press / compass and ruler]

41. How to calculate the circumference of an ellipse (Jongding Wang) 03/14/95
I am writing a program that can place character along the arch of an ellipse with a equal spacing. Thus, I have to get the circumference first. [Riemann surface]

42. Quadrilateral puzzle (Michael Thwaites) 03/09/95
If ABCD is a quadrilateral with the property that the midpoints of the sides are the four corners of a rectangle, show that AB*AB+CD*CD=BC*BC+DA*DA, that is the sums of the squares of the opposite sides are equal. [The midpoints of the quadrilateral form a parallelogram whose sides are parallel to the diagonals of the quadrilateral... / the condition originally could have been that the diagonals are perpendicular]

43. On triangle-centers determined by angles (John Conway) 03/09/95
Given three points O,H,I how many triangles are there having these points as their circumcenter, orthocenter, and incenter (respectively)?

44. Maximum angle (Gerald D. Brown) 03/03/95
Let A, B, and C be distinct collinear points with B between A and C. Let r be a ray emanating from point C that is not collinear with AC. Construct a point P on r such that angle APB is maximized.

45. Triangles on a triangle (Gerald D. Brown) 02/24/95
Let ABC be an arbitrary triangle. Let A'BC, B'AC, and C'BA be similar isosceles triangles whose bases are the sides of triangle ABC. Prove that if the three triangles are all facing outwards (or all inwards), then the lines A'A, B'B, and C'C are concurrent. [generalization of Napoleon's Theorem / a further generalization - Mathematical Gazette / J. F. Rigby]

46. Nine-point circle (Scott R Edgell) 02/13/95
Let ABC be a triangle and R and r the radii of its circumcircle and innercircle respectively. Show that R >= 2r or saying it another way R-2r >=0. [Conway: Here's a nice proof. The 9-point circle has radius R/2. But it sticks out from each side of the triangle, so must have radius at least that of the incircle, which just touches.]

47. Medians of a rt. triangle... (Rinobullet) 02/10/95
The medians of a right triangle which are drawn from the vertices of the acute angles are 5 and sq root of 40. What is the length of the hypotenuse?

48. A puzzle and a meta-puzzle (Erann Gat) 02/09/95
The puzzle: Prove that sqrt(3)/sin(20) - 1/cos(20) = 4. The meta-puzzle: What is the relationship of this puzzle with the isoscoles triangle "grad school" puzzle? [ I used the first 5 lines above in my proof that the tan(
49. Triangle Division (Gerald D. Brown) 02/09/95
Given: An arbitrary triangle and an arbitrary point ( inside, on, or outside the triangle ). Construct: A line through the point that divides the triangle into two polygons of equal area.

50. Three circles (Rinobullet) 02/08/95
There are three circles ( A, B, and C) with centers (a, b, and c) and each circle is tangent to the other two. If ab=10, ac=15, and bc=20, then what is the radius of circle C? [four configurations of internal and external tangency / a little substitution and some algebra]

51. Partitioning equal-area rectangles? (Bill Margolis) 02/08/95
How do I partition a unit area rectangle of width x and height 1/x, say, where x = 1 + pi/100, into at most 4 disjoint triangles, which can then be reassembled into a unit square? (Or any finite number of triangles, for that matter). [Tarski / ASCII diagrams / When Hilbert was investigating the foundations of geometry, he went to the trouble of proving that given any two polygons of the same area one could dissect one into a finite number of polygonal pieces that could be reassembled to form the other / Quick proof]

52. Grad School Puzzle (what does angle DEB equal?) (John Conway) 01/31/95
Draw an isosceles triangle ABC where angle B & C are the same (80 degrees). Place a point somewhere on side AB and call it D. Place a point on side AC and call it E. Connect segment BE, DE, and CD. We know this much.... angle EBC = 60, angle EBD = 20, angle DCE = 30, angle DCB = 50. The question is what does angle DEB =?

53. Venn diagrams (Delbecque Yannick) 01/21/95
I know there is 2^n-1 disjoints set (without the empty set) generated by n set (supposing that intersections are not empty). We usually represent the situation with a Venn diagram, the 2^-1 sets corresponding to 2^n-1 areas in the diagram. I'm sure all of us have seen those diagrams for cases n=2 and n=3. I recently discovered one for n=4, which represented each set with one region (and only one, since it is possible draw diagrams where sets get split by others... Since then, I've tried to construct an equivallent diagram for n=5, without success. After a little thinking, I now conjecture that there is no such representation. I've not been able to prove it (nor to get any counter example by friends or in math literature...). Can someone here solve this problem? [It IS actually possible to draw Venn diagrams for n generic sets for all integers n. / example for 4 / it could be possible to do it with only straiths in