Page 12 - Science Focus (Issue 018)

(1729 = 1 + 12 = 9 + 10 ). The rest of the taxicab numbers
3
3
3
3
are defined similarly – the nth taxicab number is the
smallest number that can be written as the sum of n
distinct pairs of positive cubes [5].
To end this article, here is a little problem, related to
continued fractions, for you to solve:

“A street has between 50 and 500 houses in a row, numbered 1,
2, 3, 4, … consecutively. There is a certain house on the street
such that the sum of all the house numbers to the left side
of it is equal to the sum of all the house numbers to its right. 就讓我以一條與連分數相關的小問題作結吧，你能解
Find the number of this house and the total number of houses 答嗎？
on the street. [6]”
The story behind [7]: One of Ramanujan’s
contemporaries at Cambridge, P. C. Mahalanobis, ˜ίɓૢ൑ɪdϞɓΐᅰͦމ Ї ගٙג܊dג܊໮
read about this problem while visiting Ramanujan. ᇁ஗නҏᇜϓ e e e nn ί൑ɪݔɓࢭגɿd̴̸ᗙהϞ
After solving the problem, Mahalanobis turned to ג܊໮ᇁٙᐼձd༧̛ᗙ໮ᇁٙᐼձ݊ɓᅵٙfՑֵவࢭג
ɿٙ໮ᇁձ൑ɪٙג܊ᐼᅰ݊ޟჿk < > ™
Ramanujan, who was cooking, and read the problem
to him. While still stirring the vegetables in his pot,
வૢᕚͦߠܝϞഹνϤٙɓࡈ݂ԫ < >jז৵рږίᄏ
Ramanujan gave the answer right away – a continued ዗ٙ؃ʾ৵۞זፕˢ౶€1 $ .BIBMBOPCJTίઞૐ˼ࣛᛘ
fraction that expressed all the solutions to the problem, Ցவ༸ਪᕚf৵۞זፕˢ౶Іʉྒྷ༊༆ഈܝdکᔷԒҪਪᕚࣦ
ignoring the constraint of 50 to 500 houses. So as a ᛘഗ͍ίਂൄٙז৵рږᛓfז৵рږᒔί᛻ਗᒢʕᇴൄʘ
bonus problem, can you find the continued fraction that ყdఱͭՍ༸̈əഈࣩj˼ഗٙ݊ɓࡈঐቇ͜׵הϞઋرٙஹ
Ramanujan answered? ʱᅰ€ޟЇ̙˸ׁଫ൑༸ɪϞ Ց ගגɿவࡈࠢՓfЪ
މɓ༸̋ʱᕚdӺ௞ז৵рږഗٙஹʱᅰ݊ޟჿճk

Answer: House number 204, on a street with 288
houses. In general, the continued fraction ഈࣩj204 號房屋、街上共有 288 間房子。簡單來說，
連分數

gives successively the fractions
能給出以下的分數：
The number of houses n and the house number x are
expressed by the fractions as . A full explanation 街上房屋總數n 和該房屋的號碼x 可以在那些分數中被表
can be found in reference [7]. 示為 。 詳解可於參考資料 [7] 找到。

References 參考資料：
[1] O'Connor, J. J., & Robertson, E. F. (1998). Srinivasa Aiyangar Ramanujan. Retrieved from https://mathshistory.st-andrews.ac.uk/
Biographies/Ramanujan/
[2] Mastin, L. (n.d.). G. H. Hardy: Ramanujan's Mentor. Retrieved from https://www.storyofmathematics.com/20th_hardy.html
[3] Hardy, G. H. (1937). The Indian Mathematician Ramanujan. The American Mathematical Monthly, 44(3), 137-155.
[4] Andrews, G. E. (2012). The Discovery of Ramanujan’s Lost Notebook. Retrieved from https://faculty.math.illinois.edu/~berndt/
andrews.pdf
[5] Weisstein, E. W. (n.d.). Taxicab Number. Retrieved from https://mathworld.wolfram.com/TaxicabNumber.html
[6] Brilliant.org. (n.d.). Srinivasa Ramanujan. Retrieved from https://brilliant.org/wiki/srinivasa-ramanujan/
[7] Mutalik, P. (2016, August 8). Solution: 'Puzzles Inspired by Ramanujan'. Retrieved from https://www.quantamagazine.org/
solution-puzzles-inspired-by-ramanujan-20160808