3.1: Twenty problems which embody “3 — 1 = 2”
- Page ID
- 23450
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The answer to every one of the questions in Problem 78 is the same - at least, as a ‘pure number’. The goal is therefore not to “solve” each problem, but to distinguish between, and to reflect upon, the different ways in which the very simple mathematical structure “3 — 1 = 2” turns out to be the relevant “model” in each case.
Problem 78
(a) I was given three apples, and then ate two of them. How many were left?
(b) A barge-pole three metres long stands upright on the bottom of the canal, with one metre protruding above the surface. How deep is the water in the canal?
(c) Tanya said: “I have three more brothers than sisters”. How many more boys are there in Tanya’s family than girls?
(d) How many cuts do you have to make to saw a log into three pieces?
(e) A train was due to arrive one hour ago. We are told that it is three hours late. When can we expect it to arrive?
(f) A brick and a spade weigh the same as three bricks. What is the weight of the spade?
(g) The distance between each successive pair of milestones is 1 mile. I walk from the first milestone to the third one. How far do I walk?
(h) The arithmetic mean (or average) of two numbers is 3. If half their difference is 1, what is the smaller number?
(i) The distance from our house to the train station is 3 km. The distance from our house to Mihnukhin’s house along the same road is 1 km. What is the distance from the station to Mihnukhin’s house?
(j) In one hundred years’ time we will celebrate the tercentenary of our university. How many centuries ago was it founded?
(k) In still water I can swim 3 km in three hours. In the same time a log drifts 1 km downstream in the river. How many kilometres would I be able to swim in the same time travelling upstream in the same river?
(l) December 2nd fell on a Sunday. How many working days preceded the first Tuesday of that month?4
4 This question is historically correct. In 1946, in the Soviet Union, when these problems were formulated, Saturday was a working day.
(m) I walk with a speed of 3 km per hour. My friend is some distance ahead of me, and is walking in the same direction pushing his broken down motorbike at 1 km per hour. At what rate is the distance between us diminishing?
(n) A trench 3 km long was dug in a week by three crews of diggers, all working at the same rate as each other. How many such crews would be needed to dig a trench 1 km shorter in the same time?
(o) Moscow and Gorky are cities in adjacent time zones. What is the time in Moscow when it is 3 pm in Gorky?5
(p) An old ‘rule-of-thumb’ for anti-aircraft gunners stated that: To hit a plane from a stationary anti-aircraft gun, one should aim at a point exactly three plane’s lengths ahead of the moving plane. Now suppose that the gun was actually moving in the same direction as the plane with one third of the plane’s speed. At what point should the gunner aim his fire?
(q) My brother is three times as old as I am. How many times my present age was his age when I was born?
(r) I add 1 to a number and the result is a multiple of 3. What would the remainder be if I were to divide the original number by 3?
(s) It takes 1 minute for a train 1 km long to completely pass a telegraph pole by the track side. At the same speed the train passes right through a tunnel in 3 minutes. What is the length of the tunnel?
(t) Three trams operate on a two-track route, with trams travelling in one direction on one track and returning on the other track. Each tram remains a fixed distance of 3 km behind the tram in front. At a particular moment one tram is exactly 1 km away from the tram on the opposite track. How far is the third tram from its nearest neighbour?