Question Number 211614 by hardmath last updated on 14/Sep/24
$$ \\ $$A group of people are standing in a circle. Every second person is removed from the circle and this process continues until only one person remains in the circle. If there are 100 people in the circle, what will be the number of the last person left?
Commented by nikif99 last updated on 15/Sep/24
Answered by nikif99 last updated on 15/Sep/24
$${Person}\:\mathrm{73}.\:{I}\:{give}\:{a}\:{graphic}\:{solution}. \\ $$$${Initially}\:{all}\:{persons}\:{are}\:{numbered}\:\mathrm{1}−\mathrm{100}. \\ $$$${In}\:{the}\:\mathrm{1}{st}\:{round}\:\left({vertically}\right)\:{all}\:{even} \\ $$$${numbers}\:{are}\:{removed},\:{while}\:{odd}\: \\ $$$${numbers}\:{are}\:{ranked}\:{again}\:\mathrm{1}−\mathrm{50}. \\ $$$${In}\:{the}\:\mathrm{2}{nd}\:{round}\:\left({next}\:{column}\right), \\ $$$${person}\:\mathrm{1}\:{is}\:{staying}\:{while}\:{all}\:{even} \\ $$$${persons}\:{that}\:{follows}\:{are}\:{gone}, \\ $$$$…{and}\:{so}\:{on}\:{till}\:{the}\:{last}\:{round}. \\ $$$$\underline{{How}\:{to}\:{read}\:{the}\:{graph}:} \\ $$$$\boldsymbol{{a}}.\:{if}\:{the}\:{last}\:{person}\:{of}\:{a}\:{round}\:{is} \\ $$$${removing},\:{it}\:{is}\:{ranked}\:\mathrm{0},\:{marked}\:{in} \\ $$$${light}\:{gray},\:{so}\:{person}\:\mathrm{1}\:{in}\:{the}\:{next} \\ $$$${round}\:{stays}. \\ $$$$\boldsymbol{{b}}.\:{if}\:{the}\:{last}\:{person}\:{is}\:{staying},\:{it}\:{is} \\ $$$${ranked}\:{with}>\mathrm{0}\:{so}\:{person}\:\mathrm{1}\:{in}\:{the}\:{next} \\ $$$${round}\:{is}\:{ranked}\:\mathrm{0}\:{and}\:{gone}. \\ $$$$\boldsymbol{{c}}.\:{all}\:{previously}\:{removed}\:{persons}\:{are} \\ $$$${ranked}\:−\mathrm{1}\:{and}\:{marked}\:{in}\:{dark}\:{gray}. \\ $$
Commented by hardmath last updated on 15/Sep/24
$$\mathrm{dear}\:\mathrm{professor}\:\mathrm{thank}\:\mathrm{you} \\ $$$$\mathrm{answer}:\:\mathrm{73}\:? \\ $$
Commented by nikif99 last updated on 15/Sep/24
$${according}\:{to}\:{calc}\:{and}\:{algorithm}, \\ $$$${person}\:\mathrm{73}\:{is}\:{the}\:{last}\:{one}. \\ $$