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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?
$$ \\ $$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 73. I give a graphic solution.  Initially all persons are numbered 1−100.  In the 1st round (vertically) all even  numbers are removed, while odd   numbers are ranked again 1−50.  In the 2nd round (next column),  person 1 is staying while all even  persons that follows are gone,  ...and so on till the last round.  How to read the graph:  a. if the last person of a round is  removing, it is ranked 0, marked in  light gray, so person 1 in the next  round stays.  b. if the last person is staying, it is  ranked with>0 so person 1 in the next  round is ranked 0 and gone.  c. all previously removed persons are  ranked −1 and marked in dark gray.
$${Person}\:\mathrm{73}.\:{I}\:{give}\:{a}\:{graphic}\:{solution}. \\ $$$${Initially}\:{all}\:{persons}\:{are}\:{numbered}\:\mathrm{1}−\mathrm{100}. \\ $$$${In}\:{the}\:\mathrm{1}{st}\:{round}\:\left({vertically}\right)\:{all}\:{even} \\ $$$${numbers}\:{are}\:{removed},\:{while}\:{odd}\: \\ $$$${numbers}\:{are}\:{ranked}\:{again}\:\mathrm{1}−\mathrm{50}. \\ $$$${In}\:{the}\:\mathrm{2}{nd}\:{round}\:\left({next}\:{column}\right), \\ $$$${person}\:\mathrm{1}\:{is}\:{staying}\:{while}\:{all}\:{even} \\ $$$${persons}\:{that}\:{follows}\:{are}\:{gone}, \\ $$$$…{and}\:{so}\:{on}\:{till}\:{the}\:{last}\:{round}. \\ $$$$\underline{{How}\:{to}\:{read}\:{the}\:{graph}:} \\ $$$$\boldsymbol{{a}}.\:{if}\:{the}\:{last}\:{person}\:{of}\:{a}\:{round}\:{is} \\ $$$${removing},\:{it}\:{is}\:{ranked}\:\mathrm{0},\:{marked}\:{in} \\ $$$${light}\:{gray},\:{so}\:{person}\:\mathrm{1}\:{in}\:{the}\:{next} \\ $$$${round}\:{stays}. \\ $$$$\boldsymbol{{b}}.\:{if}\:{the}\:{last}\:{person}\:{is}\:{staying},\:{it}\:{is} \\ $$$${ranked}\:{with}>\mathrm{0}\:{so}\:{person}\:\mathrm{1}\:{in}\:{the}\:{next} \\ $$$${round}\:{is}\:{ranked}\:\mathrm{0}\:{and}\:{gone}. \\ $$$$\boldsymbol{{c}}.\:{all}\:{previously}\:{removed}\:{persons}\:{are} \\ $$$${ranked}\:−\mathrm{1}\:{and}\:{marked}\:{in}\:{dark}\:{gray}. \\ $$
Commented by hardmath last updated on 15/Sep/24
dear professor thank you  answer: 73 ?
$$\mathrm{dear}\:\mathrm{professor}\:\mathrm{thank}\:\mathrm{you} \\ $$$$\mathrm{answer}:\:\mathrm{73}\:? \\ $$
Commented by nikif99 last updated on 15/Sep/24
according to calc and algorithm,  person 73 is the last one.
$${according}\:{to}\:{calc}\:{and}\:{algorithm}, \\ $$$${person}\:\mathrm{73}\:{is}\:{the}\:{last}\:{one}. \\ $$
Answered by mr W last updated on 19/Sep/24
say in the first round the even   numbers are removed (red numbers)  1/2/3/4/.../98/99/100/1...  1/3/5/7/.../95/97/99/1...  1/5/9/13/.../89/93/97/1...  1/9/17/25/.../81/89/97/9...  9/25/41/57/73/89/9...  9/41/73/9...  9/73  ⇒number 73 is the last number
$${say}\:{in}\:{the}\:{first}\:{round}\:{the}\:{even}\: \\ $$$${numbers}\:{are}\:{removed}\:\left({red}\:{numbers}\right) \\ $$$$\mathrm{1}/\mathrm{2}/\mathrm{3}/\mathrm{4}/…/\mathrm{98}/\mathrm{99}/\mathrm{100}/\mathrm{1}… \\ $$$$\mathrm{1}/\mathrm{3}/\mathrm{5}/\mathrm{7}/…/\mathrm{95}/\mathrm{97}/\mathrm{99}/\mathrm{1}… \\ $$$$\mathrm{1}/\mathrm{5}/\mathrm{9}/\mathrm{13}/…/\mathrm{89}/\mathrm{93}/\mathrm{97}/\mathrm{1}… \\ $$$$\mathrm{1}/\mathrm{9}/\mathrm{17}/\mathrm{25}/…/\mathrm{81}/\mathrm{89}/\mathrm{97}/\mathrm{9}… \\ $$$$\mathrm{9}/\mathrm{25}/\mathrm{41}/\mathrm{57}/\mathrm{73}/\mathrm{89}/\mathrm{9}… \\ $$$$\mathrm{9}/\mathrm{41}/\mathrm{73}/\mathrm{9}… \\ $$$$\mathrm{9}/\mathrm{73} \\ $$$$\Rightarrow{number}\:\mathrm{73}\:{is}\:{the}\:{last}\:{number} \\ $$

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