This Hardy–Ramanujan number, known as the second "taxicab number" (Ta(2)), can be expressed as 1I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."[1]

*Taxicab no. 1729 of the Hardy-Ramanujan Cab Company. (Source image via Wikimedia Commons.)*

Today, taxicabs are being replaced by ride services such as Uber, but people still drive their personal automobiles to work. In some cases, employees living near each other and working for the same company will form a carpool in which the driving duties are cycled among the participants. In a five person carpool, this would require each participant to drive on a particular weekday. This is an equitable arrangement when all these employees travel together each day, but there are invariably business trips, vacations, illness, and other circumstances, that prevent certain of them from participating on some days. This leads to the problem of who will drive on those days when the designated driver is away; and, how would this absent driver repay his colleagues for his lack of service on those days. Perhaps motivated by the US energy crisis of 1979, a mathematician and a computer scientist from IBM published a paper on carpool scheduling in 1983. I wrote about their fair carpool algorithm in an earlier article (Fair Carpooling Algorithm, April 21, 2011). The fair carpooling algorithm involves a balance sheet among the participants in which driving is considered to have a fixed cost. To keep accounting in the integer realm, this cost

*A New York City Subway token. Subway tokens were replaced by the more manageable Metrocard on April 14, 2003.Every time I see one of these, I think of Y Combinator.(Wikimedia Commons image by Jessamyn West.)*

When we were being taught derivatives in high school calculus, one of my fellow students posed a question that seemed like a rate problem. His question was whether he would be at his bus stop on time if the bus was first visible to him when he was at a certain distance from the bus stop. Since we lived in a city, the problem was that buildings will interfere with the line-of-sight, so the bus was only visible when he was quite near the bus stop. The problem is best explained by the diagram, below.

*The bus stop problem.The student is at *

Since a high school student's hormone-addled brain needs maximal teaching, we continued with derivatives and never solved the problem. The problem always stayed on my mind, so I decided to solve it. Simple geometry and a spreadsheet was all that was required. Using the property of similar triangles, we find that

Sinced/(c+a) = b/a

The following graph illustrates the results of the spreadsheet calculation using Gnumeric.t_{a}= (a+b)/r_{a}

t_{b}= (d-b)/r_{b}

*Solution of the bus stop problem as the time for the student and bus to reach the bus stop as a function of the student's distance a to the intersection.If the student sees his bus when he's more than five feet from the intersection, there's no hope that he will reach the bus stop on time.(Graphed using Gnumeric.)*

As can be deduced from the graph, the student will miss his bus if he sees it when he's more than five feet from the intersection. Just after our high school graduation, the The Hollies released the 1966 recording, Bus Stop.[5]

- Quotations of Godfrey Harold Hardy on Wikiquote.
- Ronald Fagin and John H. Williams, "A Fair Carpool Scheduling Algorithm," IBM Journal of Research and Development, vol. 27, no 2 (March 1983), pp. 133-139, DOI: 10.1147/rd.272.0133. A PDF file of this paper is available here and here.
- Jizhe Xia, Kevin M. Curtin, Weihong Li, and Yonglong Zhao, "A New Model for a Carpool Matching Service," PLOS, vol. 10, no. 6 (June 30, 2015), article no. e0129257, https://doi.org/10.1371/journal.pone.0129257. This is an open access article with a PDF file available here.
- Ed Blazina, "'GPS for transit': More real-time transit information available in Pittsburgh," Pittsburgh Post-Gazette, Aug 27, 2018.
- Bus Stop- The Hollies - 1966, YouTube Video by 74sodapop.