Sources page biographical material

6.AF. WHAT COLOUR WAS THE BEAR?
A hunter goes 100 mi south, 100 mi east and 100 mi north and finds himself where he started. He then shoots a bear -- what colour was the bear?

Square versions: Perelman; Klamkin, Breault & Schwarz; Kakinuma, Barwell & Collins; Singmaster.

I include other polar problems here. See also 10.K for related geographical problems.
"A Lover of the Mathematics." A Mathematical Miscellany in Four Parts. 2nd ed., S. Fuller, Dublin, 1735. The First Part is: An Essay towards the Probable Solution of the Forty five Surprising PARADOXES, in GORDON's Geography, so the following must have appeared in Gordon. Part I, no. 10, p. 9. "There is a particular Place of the Earth where the Winds (tho' frequently veering round the Compas) do always blow from the North Point."

Philip Breslaw (attrib.). Breslaw's Last Legacy; or the Magical Companion: containing all that is Curious, Pleasing, Entertaining and Comical; selected From the most celebrated Masters of Deception: As well with Slight of Hand, As with Mathematical Inventions. Wherein is displayed The Mode and Manner of deceiving the Eye; as practised by those celebrated Masters of Mirthful Deceptions. Including the various Exhibitions of those wonderful Artists, Breslaw, Sieur, Comus, Jonas, &c. Also the Interpretation of Dreams, Signification of Moles, Palmestry, &c. The whole forming A Book of real Knowledge in the Art of Conjuration. (T. Moore, London, 1784, 120pp.) With an accurate Description of the Method how to make The Air Balloon, and inject the Inflammable Air. (2nd ed., T. Moore, London, 1784, 132pp; 5th ed., W. Lane, London, 1791, 132pp.) A New Edition, with great Additions and Improvements. (W. Lane, London, 1795, 144pp.) Facsimile from the copy in the Byron Walker Collection, with added Introduction, etc., Stevens Magic Emporium, Wichita, Kansas, 1997. [This was first published in 1784, after Breslaw's death, so it is unlikely that he had anything to do with the book. There were versions in 1784, 1791, 1792, 1793, 1794, 1795, 1800, 1806, c1809, c1810, 1811, 1824. Hall, BCB 39-43, 46-51. Toole Stott 120-131, 966 967. Heyl 35-41. This book went through many variations of subtitle and contents -- the above is the largest version.]. I will cite the date as 1784?.

Geographical Paradoxes.

Paradox I, p. 35. Where is it noon every half hour? Answer: At the North Pole in Summer, when the sun is due south all day long, so it is noon every moment!

Paradox II, p. 36. Where can the sun and the full moon rise at the same time in the same direction? Answer: "Under the North Pole, the sun and the full moon, both decreasing in south declination, may rise in the equinoxial points at the same time; and under the North Pole, there is no other point of compass but south." I think this means at the North Pole at the equinox.

Carlile. Collection. 1793. Prob. CXVI, p. 69. Where does the wind always blow from the north?

Jackson. Rational Amusement. 1821. Geographical Paradoxes.

No. 7, pp. 36 & 103. Where do all winds blow from the north?

No. 8, pp. 36 & 110. Two places 100 miles apart, and the travelling directions are to go 50 miles north and 50 miles south.

Mr. X [cf 4.A.1]. His Pages. The Royal Magazine 10:3 (Jul 1903) 246-247. A safe catch. Airship starts at the North Pole, goes south for seven days, then west for seven days. Which way must it go to get back to its starting point? No solution given.

Pearson. 1907.

Part II, no. 21: By the compass, pp. 18 & 190. Start at North Pole and go 20 miles southwest. What direction gets back to the Pole the quickest? Answer notes that it is hard to go southwest from the Pole!

Part II, no. 15: Ask "Where's the north?" -- Pope, pp. 117 & 194. Start 1200 miles from the North Pole and go 20 mph due north by the compass. How long will it take to get to the Pole? Answer is that you never get there -- you get to the North Magnetic Pole.

Ackermann. 1925. P. 116. Man at North Pole goes 20 miles south and 30 miles west. How far, and in what direction, is he from the Pole?

Richard Guy (letter of 27 Feb 1985) recalls this problem (I think he is referring to the 'What colour was the bear' version) from his schooldays in the 1920s.

H. Phillips. Week End. 1932. Prob. 8, pp. 12 & 188. = his Playtime Omnibus, 1933, prob. 10: Popoff, pp. 54 & 237. House with four sides facing south.

H. Phillips. The Playtime Omnibus. Faber & Faber, London, 1933. Section XVI, prob. 11: Polar conundrum, pp. 51 & 234. Start at the North Pole, go 40 miles South, then 30 miles West. How far are you from the Pole. Answer: "Forty miles. (NOT thirty, as one is tempted to suggest.)" Thirty appears to be a slip for fifty??

Perelman. FFF. 1934. 1957: prob. 6, pp. 14-15 & 19-20: A dirigible's flight; 1979: prob. 7, pp. 18-19 & 25-27: A helicopter's flight. MCBF: prob. 7, pp. 18-19 & 25-26: A helicopter's flight. Dirigible/helicopter starts at Leningrad and goes 500km N, 500km  E, 500km S, 500km W. Where does it land? Cf Klamkin et seq., below.

Phillips. Brush. 1936. Prob. A.1: A stroll at the pole, pp. 1 & 73. Eskimo living at North Pole goes 3 mi south and 4 mi east. How far is he from home?

Haldeman-Julius. 1937. No. 51: North Pole problem, pp. 8 & 23. Airplane starts at North Pole, goes 30 miles south, then 40 miles west. How far is he from the Pole?

J. R. Evans. The Junior Week End Book. Gollancz, London, 1939. Prob. 9, pp. 262 & 268. House with four sides facing south.

Leopold. At Ease! 1943. A helluva question!, pp. 10 & 196. Hunter goes 10 mi south, 10 mi west, shoots a bear and drags it 10 mi back to his starting point. What colour was the bear? Says the only geographic answer is the North Pole.

E. P. Northrop. Riddles in Mathematics. 1944. 1944: 5-6; 1945: 5-6; 1961: 15 16. He starts with the house which faces south on all sides. Then he has a hunter that sees a bear 100 yards east. The hunter runs 100 yards north and shoots south at the bear -- what colour .... He then gives the three sided walk version, but doesn't specify the solution.

E. J. Moulton. A speed test question; a problem in geography. AMM 51 (1944) 216 & 220. Discusses all solutions of the three-sided walk problem.

W. A. Bagley. Puzzle Pie. Op. cit. in 5.D.5. 1944. No. 50: A fine outlook, pp. 54-55. House facing south on all sides used by an artist painting bears!

Leeming. 1946. Chap. 3, prob. 32: What color was the bear?, pp. 33 & 160. Man walks 10 miles south, then 10 miles west, where he shoots a bear. He drags it 10 miles north to his base. What color .... He gives only one solution.

Darwin A. Hindman. Handbook of Indoor Games & Contests. (Prentice Hall, 1955); Nicholas Kaye, London, 1957. Chap. 16, prob. 4: The bear hunter, pp. 256 & 261. Hunter surprises bear. Hunter runs 200 yards north, bear runs 200 yards east, hunter fires south at bear. What colour ....

Murray S. Klamkin, proposer; D. A. Breault & Benjamin L. Schwarz, solvers. Problem 369. MM 32 (1958/59) 220 & 33 (1959/60) 110 & 226 228. Explorer goes 100 miles north, then east, then south, then west, and is back at his starting point. Breault gives only the obvious solution. Schwartz gives all solutions, but not explicitly. Cf Perelman, 1934.

Benjamin L. Schwartz. What color was the bear?. MM 34 (1960) 1-4. ??NYS -- described by Gardner, SA (May 1966) = Carnival, chap. 17. Considers the problem where the hunter looks south and sees a bear 100 yards away. The bear goes 100 yards east and the hunter shoots it by aiming due south. This gives two extra types of solution.

Ripley's Puzzles and Games. 1966. Pp. 18, item 5. 50 mi N, 1000 mi W, 10 mi S to return to your starting point. Answer only gives the South Pole, ignoring the infinitely many cases near the North Pole. Looking at this made me realise that when the sideways distance is larger than the circumference of the parallel at that distance from the pole, then there are other solutions that start near the pole. Here there are three solutions where one starts at distances 109.2, 29.6 or 3.05 miles from the South Pole, circling it 1, 2 or 3 times.

Yasuo Kakinuma, proposer; Brian Barwell and Craig H. Collins, solvers. Problem 1212 -- Variation of the polar bear problem. JRM 15:3 (1982 83) 222 & 16:3 (1983-84) 226 228. Square problem going one mile south, east, north, west. Barwell gets the explicit quadratic equation, but then approximates its solutions. Collins assumes the earth is flat near the pole.

David Singmaster. Bear hunting problems. Submitted to MM, 1986. Finds explicit solutions for the general version of Perelman/Klamkin's problem. [In fact, I was ignorant of (or had long forgotten) the above when I remembered and solved the problem. My thanks to an editor (Paul Bateman ??check) for referring me to Klamkin. The Kakinuma et al then turned up also.] Analysis of the solutions leads to some variations, including the following.

David Singmaster. Home is the hunter. Man heads north, goes ten miles, has lunch, heads north, goes ten miles and finds himself where he started.

Used as: Explorer's problem by Keith Devlin in his Micromaths Column; The Guardian (18 Jun 1987) 16 & (2 Jul 1987) 16.

Used by me as one of: Spring term puzzles; South Bank Polytechnic Computer Services Department Newsletter (Spring 1989) unpaged [p. 15].

Used by Will Shortz in his National Public Radio program 6? Jan 1991.

Used as: A walk on the wild side, Games 15:2 (No. 104) (Aug 1991) 57 & 40.

Used as: The hunting game, Focus 3 (Feb 1993) 77 & 98.

Used in my Puzzle Box column, G&P, No. 11 (Feb 1995) 19 & No. 12 (Mar 1995) 41.

Bob Stanton. The explorers. Games Magazine 17:1 (No. 113) (Feb 1993) 61 & 43. Two explorers set out and go 500 miles in each direction. Madge goes N, W, S, E, while Ellen goes E, S, W, N. At the end, they meet at the same point. However, this is not at their starting point. How come? and how far are they from their starting point, and in what direction? They are not near either pole.

Yuri B. Chernyak & Robert S. Rose. The Chicken from Minsk. BasicBooks, NY, 1995. Chap. 11, prob. 9: What color was that bear? (A lesson in non-Euclidean geometry), pp. 97 & 185-191. Camper walks south 2 km, then west 5 km, then north 2 km; how far is he from his starting point? Solution analyses this and related problems, finding that the distance x satisfies 0  x  7.183, noting that there are many minimal cases near the south pole and if one is between them, one gets a local maximum, so one has to determine one's position very carefully.

David Singmaster. Symmetry saves the solution. IN: Alfred S. Posamentier & Wolfgang Schulz, eds.; The Art of Problem Solving: A Resource for the Mathematics Teacher; Corwin Press, NY, 1996, pp. 273-286. Sketches the explicit solution to Klamkin's problem as an example of the use of symmetric variables to obtain a solution.

Anonymous. Brainteaser B163 -- Shady matters. Quantum 6:3 (Jan/Feb 1996) 15 & 48. Is there anywhere on earth where one's shadow has the same length all day long?