I-have-25-horses-and-I-d-like-to-know-which-are-the-three-fastest-horses-among-them-I-do-not-have-a-clock-but-I-have-a-race-track-which-can-be-used-by-5-horses-at-a-time-If-each-horse-covers-the-dis

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Question Number 3274 by Yozzi last updated on 09/Dec/15

$$Ihave25horsesand{I}^{\prime }dliketo$$$$knowwhicharethethreefastest$$$$horsesamongthem.Idonothavea$$$$clockbutIhavearacetrackwhich$$$$canbeusedby5horsesatatime.$$$$Ifeachhorsecoversthedistance$$$$ofthetrackinthesametimefor$$$$everyraceitruns,findtheleast$$$$numberofracesthatoughttobe$$$$heldinordertoidentifythethree$$$$fastesthorses.$$$$$$

Answered by prakash jain last updated on 09/Dec/15

$$\mathrm{First}5\mathrm{races}\mathrm{with}\mathrm{numbered}\mathrm{results}$$$$1:{\mathrm{h}}_1{\mathrm{h}}_2{\mathrm{h}}_3{\mathrm{h}}_4{\mathrm{h}}_5$$$$2:{\mathrm{h}}_6{\mathrm{h}}_7{\mathrm{h}}_8{\mathrm{h}}_9{\mathrm{h}}_{10}$$$$3:{\mathrm{h}}_{11}{\mathrm{h}}_{12}{\mathrm{h}}_{13}{\mathrm{h}}_{14}{\mathrm{h}}_{15}$$$$4:{\mathrm{h}}_{16}{\mathrm{h}}_{17}{\mathrm{h}}_{18}{\mathrm{h}}_{19}{\mathrm{h}}_{20}$$$$5:{\mathrm{h}}_{21}{\mathrm{h}}_{22}{\mathrm{h}}_{22}{\mathrm{h}}_{24}{\mathrm{h}}_{25}$$$$2\mathrm{elimnated}\mathrm{from}\mathrm{each}\mathrm{group}\mathrm{since}\mathrm{they}\mathrm{cannot}$$$$\mathrm{take}\mathrm{position}123\mathrm{overall}.$$$$6\mathrm{th}\mathrm{Race}\mathrm{with}\mathrm{toppers}({\mathrm{h}}_1,{\mathrm{h}}_6,{\mathrm{h}}_{11},{\mathrm{h}}_{16},{\mathrm{h}}_{21})$$$$\mathrm{Let}\mathrm{us}\mathrm{assume}\mathrm{results}\mathrm{are}$$$${\mathrm{h}}_1{\mathrm{h}}_6{\mathrm{h}}_{16}{\mathrm{h}}_{21}{\mathrm{h}}_{11}$$$${\mathrm{h}}_{11}/{\mathrm{h}}_{21}\mathrm{is}4/5\mathrm{among}\mathrm{toppers}\mathrm{all}\mathrm{participants}$$$$\mathrm{of}\mathrm{race}3\mathrm{and}5\mathrm{are}\mathrm{eliminated}\mathrm{as}\mathrm{they}\mathrm{are}$$$$\mathrm{slower}\mathrm{than}\mathrm{at}\mathrm{least}{4}^{th}\mathrm{position}$$$$\mathrm{Leave}\mathrm{selection}\mathrm{process}\mathrm{limited}\mathrm{to}$$$${\mathrm{h}}_1{\mathrm{h}}_6{\mathrm{h}}_{16}{\mathrm{h}}_2{\mathrm{h}}_3{\mathrm{h}}_7{\mathrm{h}}_8{\mathrm{h}}_{17}{\mathrm{h}}_{18}$$$${\mathrm{h}}_{16}\mathrm{is}\mathrm{at}\mathrm{best}{3}^{\mathrm{rd}}\mathrm{so}{\mathrm{h}}_{17}\mathrm{and}{\mathrm{h}}_{18}\mathrm{eliminated}.$$$${\mathrm{h}}_1{\mathrm{h}}_6{\mathrm{h}}_{16}{\mathrm{h}}_2{\mathrm{h}}_3{\mathrm{h}}_7{\mathrm{h}}_8$$$${\mathrm{h}}_6\mathrm{is}\mathrm{at}\mathrm{best}\mathrm{second}\mathrm{so}{\mathrm{h}}_8\mathrm{can}\mathrm{be}\mathrm{at}\mathrm{best}{4}^{\mathrm{th}}.$$$${\mathrm{h}}_1{\mathrm{h}}_6{\mathrm{h}}_{16}{\mathrm{h}}_2{\mathrm{h}}_3{\mathrm{h}}_7$$$$\mathrm{Leaves}6.$$$${\mathrm{h}}_1\mathrm{is}\mathrm{first}\mathrm{so}\mathrm{it}\mathrm{is}\mathrm{clearly}\mathrm{the}\mathrm{first}.\mathrm{need}\mathrm{to}$$$$\mathrm{find}\mathrm{best}2\mathrm{from}\mathrm{remainin}5$$$${\mathrm{h}}_6{\mathrm{h}}_{16}{\mathrm{h}}_2{\mathrm{h}}_3{\mathrm{h}}_7$$$$\mathrm{Race}7\mathrm{to}\mathrm{determine}\mathrm{the}{2}^{\mathrm{nd}}\mathrm{and}{3}^{\mathrm{rd}}$$$$7\mathrm{Races}\mathrm{are}\mathrm{needed}.$$

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