Snowball Problems Melted

The snow from the record setting snowfalls in Boston are pretty much gone, so here are solutions to the two snowball problems from the February.

Solution to Snowball Problem #1.

We claim that every point in space on the side of the plane with the rolling snowball and within a snowball diameter of the plane can be visited by the marked point.

Let C be the circumference of the snowball. Any positive distance less than 2C can serve as the base of an isosceles triangle with equal sides of length C. Draw this triangle in the plane and put the snowball at one endpoint of the base with its marked point touching the plane. Roll the snowball along the two sides of length C. It ends up at the other endpoint of the base with its marked point touching the plane. This shows that the marked point can visit any point within a circle of radius 2C from its starting point. By taking several such journeys, we see that every point in the plane can be visited. For any point P on the same side of the plane as the snowball and within a snowball’s diameter of the plane, position the snowball so that its marked point is at P. We can roll this snowball until the marked point is touching the plane and since all points in the plane are visitable, we conclude that P can also be visited.

Solution to Snowball Problem #2 (Noah Fechtor-Pradines).

We claim that the locus of visitable points is a cardioid of revolution.

Let’s mark the point on the sphere where the marked point on the snowball touches it at the start with a golden dot. Look at the situation from the point-of-view of the common tangent plane. The two balls roll as mirror images of each other. The marked points are reflections of each other in the common tangent plane.

In other words, no matter how the snowball rolls about the sphere, the marked point will be located at the reflection of the golden dot in the common tangent.

We conclude that the locus of visitable points is a surface with rotational symmetry (about the axis one of whose poles is the golden dot). A vertical cross section by a plane containing the pole of the stationary sphere is the curve traced out by a dot on a circle as it rolls about a circle of the same diameter, and this is known to be a cardioid.

 

 

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Girl Scouts STEM Expo crossword raffle

(Added April 5, 2015: This raffle is now closed. Thank you to all who entered. Congratulations to Monica G. who won the member draw and Danielle D. who won the general draw!)

crossword gird Last Sunday, Girls’ Angle hosted a booth at the Girl Scouts STEM Expo in Framingham, Massachusetts. Visitors were challenged to accomplish 3 tasks, one of which was solving this crossword puzzle. They succeeded, so now we’re opening this puzzle to the general public. Solve this crossword alone or with friends and relatives.  Scan in or take a digital photo of your completed grid and send it to girlsanglepuzzler “at” gmail.com. We will randomly draw from the correct answers received by midnight on April 4, 2015 to select a “winner” and send the winner a small prize. Girls’ Angle members will be put into a separate pool for a different prize. (If you worked on this puzzle at the Girl Scouts STEM Expo, you can’t enter this raffle. Sorry!)

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Girls’ Angle Bulletin, Volume 8, Number 3

Cover of Girls' Angle Bulletin, Volume 8, Number 3The electronic version of the latest issue of the Girls’ Angle Bulletin is now available on our website.

Volume 8, Number 3 of the Bulletin kicks off with the first part of a 2-part interview with Hamilton College Assistant Professor of Mathematics Courtney Gibbons.  In this first half, she discusses how she got interested in math, tells us about how she learns, studies, and creates math, and explains the notions of “fixing set” and “fixing number.”  She also tells us about some of her goals as a mathematician.

The best way to learn math is to do lots of problems: after all, that’s what math is! – Courtney Gibbons

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We’ve Got Snowball Problems

blog_021815_01Current weather in New England makes it hard to not think about snow. So here are two snowball inspired math problems. We welcome your solutions (send to girlsangle “at” gmail.com). Perhaps we’ll post solutions when the last traces of snow have cleared from Boston streets.

Both problems involve a snowball in the shape of a perfect sphere. Mark a point on the surface of the snowball. The snowball rolls without slipping. Also, there is no unnecessary rotation. (In other words, the snowball rotates only around an axis perpendicular to the plane of the great circle that is tangent to the path that the snowball travels.) In each problem, the snowball starts with the marked point touching the surface it is rolling on.

Snowball Problem 1. If the snowball rolls about on a plane, what is the locus of all the points in space that the marked point can visit?

Snowball Problem 2. If the snowball rolls about on another sphere of the same radius, what is the locus of all the points in space that the marked point can visit?

 

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Happy New Year 2015!

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Girls’ Angle Bulletin, Volume 8, Number 2

Cover of Girls' Angle Bulletin, Volume 8, Number 2The electronic version of the latest issue of the Girls’ Angle Bulletin is now available on our website.

After reading Stained Glass Angles in Volume 7, Number 4, Marion Walter happened upon an amazing stained glass window by John Rose at the Eugene Public Library in Oregon. That inspired her to suggest a Math Buffet column featuring mathematically inspired stained glass windows. One thing led to another, and in this issue, we feature stained glass windows from 7 designers. The cover shows a detail of a stained glass window designed by Millie Wert, a graduate of Harpeth Hall.  (Added January 27, 2015: For more math inspired stained-glass windows by students of Thaddeus Wert at Harpet Hall, please visit his blog.) Continue reading

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Go Digital! A Crossword Puzzle Raffle

(Added January 7, 2015: This raffle is now closed. Thank you to all who entered. Congratulations to F. Dangerfield of Massachusetts who won the member draw and to B. Jackson of Wisconsin who won the general draw!)

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At Girls’ Angle, we do love digits!

Solve this crossword alone or with friends and relatives.  Scan in or take a digital photo of your completed grid and send it to girlsanglepuzzler “at” gmail.com. We will randomly draw from the correct answers received by midnight on January 4, 2015 to select a “winner” and send the winner a small prize. Girls’ Angle members will be put into a separate pool for a different prize.

This crossword puzzle is dedicated to Connie Chow.

For a print friendly version, click here.

GoDigitalGrid

Across

1. Rest
6. Fall mo.
9. Sub
14. Breathing
15. Greek letter
16. Loaded with marinara
17. Hushed
18. Goal
19. Vermeer product
20. Sunrises and hatchings
23. Soft cheese
24. It often gets beat
25. Rights wrongs
27. City in 54D
31. Donor
33. Unlike Smith
34. Acts like a ghost
36. Where cultures are made
39. Für Elise form
41. Type of health plan: Abbr.
42. Rise over run
44. Increases
45. Forest and jungle
48. Divisible by 2
49. Catches up to
50. Most frigid
52. Triangular sail, or, a 15-year-old who lives near the Endeavour?
55. Hamburg’s commission
56. Drive the getaway car
57. Equal areas in equal times
64. Burdens
66. Overly
67. Baking bean
68. Bad color combination
69. Trig function
70. Seaweeds
71. Veracity
72. Time unit’s plural form: Abbr.
73. Name with a double consonant

Down

1. Docile
2. Graduate
3. Flying diamond
4. Follows for or what
5. Pentagon property
6. Type of ape
7. Elegant
8. The Princeton Companion to Mathematics, for example
9. Hawaii is the last of these: Abbr.
10. Samson’s bane?
11. No longer useful
12. Corrosive
13. Kids
21. They can be bruised
22. Oft used file command
26. Place to eat
27. Brown
28. Continuous image of the circle
29. Writes
30. Total
31. Most exciting match
32. Fe
35. Soften in the mouth
36. Zero to Federer
37. What cosine does to sine
38. Not straight
40. Shrek
43. Hawaiian graduation gift
46. File’s partner
47. Tails
49. Loathe
51. Someone you only drive to work with?
52. Fastener
53. Haliotis
54. State of 27A
55. Clean with string
58. Draw
59. Low quality
60. Shelf with a view
61. Plastic brick
62. Not home
63. Gave more importance
65. Short

Good luck!

(We will not use your contact information for any purpose other than to deliver your prize, should you win. After the winner has been selected, all emails received will be promptly deleted. At the winner’s discretion, we will let you know who won. Anyone who makes more than one submission will be disqualified! Sorry! Also, this offer is only valid in those states in the United States where such things are legal. There is no fee to enter this puzzle contest.)

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