Sources page biographical material

5.D.2. RULER WITH MINIMAL NUMBER OF MARKS
Dudeney. Problem 518: The damaged measure. Strand Mag. (Sep 1920) ??NX. Wants a minimal ruler for 33 inches total length. (=? MP 180)

Dudeney. Problem 530: The six cottagers. Strand Mag. (Jan 1921) ??NX. Wants 6 points on a circle to give all arc distances 1, 2, ..., 20. (=? MP 181)

Percy Alexander MacMahon. The prime numbers of measurement on a scale. Proc. Camb. Philos. Soc. 21 (1922 23) 651 654. He considers the infinite case, i.e. a(0) = 0, a(i+1) = a(i) + least integer which is not yet measurable. This gives: 0, 1, 3, 7, 12, 20, 30, 44, ....

Dudeney. MP. 1926.

Prob. 180: The damaged measure, pp. 77 & 167. (= 536, prob. 453, pp. 173, 383 384.) Mark a ruler of length 33 with 8 (internal) marks. Gives 16 solutions.

Prob. 181: The six cottagers, pp. 77 78 & 167. = 536, prob. 454, pp. 174 & 384.

A. Brauer. A problem of additive number theory and its application in electrical engineering. J. Elisha Mitchell Sci. Soc. 61 (1945) 55 56. Problem arises in designing a resistance box.

Л. Редеи & А. Реньи [L. Redei & A. Ren'i (Rényi)]. О представленин чисел 1, 2, ..., N лосредством разностей [O predstavlenin chisel 1, 2, ... , N losredstvom raznosteĭ (On the representation of 1, 2, ..., N by differences)]. Мат. Сборник [Mat. Sbornik] 66 (NS 24) (1949) 385 389.

Anonymous. An unsolved problem. Eureka 11 (Jan 1949) 11 & 30. Place as few marks as possible to permit measuring integers up to n. For n = 13, an example is: 0, 1, 2, 6, 10, 13. Mentions some general results for a circle.

John Leech. On the representation of 1, 2, ..., n by differences. J. London Math. Soc. 31 (1956) 160 169. Improves Redei & Rényi's results. Gives best examples for small n.

Anon. Puzzle column: What's your potential? MTg 19 (1962) 35 & 20 (1962) 43. Problem posed in terms of transformer outputs -- can we arrange 6 outputs to give every integral voltage up through 15? Problem also asks for the general case. Solution asserts, without real proof, that the optimum occurs with 0, 1, 4, 7, 10, ..., n 11, n 8, n 5, n 2 or its complement.

T. H. O'Beirne. Puzzles and Paradoxes. OUP, 1965. Chap. 6 discusses several versions of the problem.

Gardner. SA (Jan 1965) c= Magic Numbers, chap. 6. Describes 1, 2, 6, 10 on a ruler 13 long. Says 3 marks are sufficient on 9 and 4 marks on 12 and asks for proof of the latter and for the maximum number of distances that 3 marks on 12 can produce. How can you mark a ruler 36 long? Says Dudeney, MP prob. 180, believed that 9 marks were needed for a ruler longer than 33, but Leech managed to show 8 was sufficient up to 36.

C. J. Cooke. Differences. MTg 47 (1969) 16. Says the problem in MTg 19 (1962) appears in H. L. Dorwart's The Geometry of Incidence (1966) related to perfect difference sets but with an erroneous definition which is corrected by references to H. J. Ryser's Combinatorial Mathematics. However, this doesn't prove the assertions made in MTg 20.

Jonathan Always. Puzzles for Puzzlers. Tandem, 1971. Prob. 22: Starting and stopping, pp. 18 & 66. Circular track, 1900 yards around. How can one place marker posts so every multiple of 100 yards up to 1900 can be run. Answer: at 0, 1, 3, 9, 15.

Gardner. SA (Mar 1972) = Wheels, Chap. 15.