N sheets of card in which 1, 2, 3, ...., N is written each are clockwise arranged in the small order of the number circularly.
When removing one card at a time in accordance with the following rule, it is considered which card remains at the end.
<Rule 1>
The card in which 1 is written is removed first.
<Rule 2>
When a certain card is removed, the 2nd card from the card counted clockwise and will be removed.
This operation is repeated until only one card remains.
For example, in case of N = 13, cards are removed as shown in Fig. 1, and #10 card finally remains.
(x mark expresses the removed card.)
Answer the following questions.
(1) Answer the number written to the card which remains at the end in case of N = 8.
(2) When cards are removed at the 1st round in case of N = 16, as shown in Fig. 2, eight cards remain.
The card in which 2 was written is removed next.
Answer the number written to the card which remains at the end in case of N = 16.
Moreover, answer the number written to the card which remains at the end, respectively in case of N = 32 and N = 64.
(3) When the cards in which 1, 3, and 5 are written at the 1st round are removed in case of N = 35, 32 cards remain.
The card in which 7 was written is removed next.
Answer the number written to the card which remains at the end in case of N = 35.
(4) When 36 cards are removed to the 1st round in case of N = 100, 64 cards remain.
Answer the number written to the card which remains at the end in case of N = 100.
(5) Answer the number written to the card which remains at the end in case of N = 2009.
When removing one card at a time in accordance with the following rule, it is considered which card remains at the end.
<Rule 1>
The card in which 1 is written is removed first.
<Rule 2>
When a certain card is removed, the 2nd card from the card counted clockwise and will be removed.
This operation is repeated until only one card remains.
For example, in case of N = 13, cards are removed as shown in Fig. 1, and #10 card finally remains.
(x mark expresses the removed card.)
Answer the following questions.
(1) Answer the number written to the card which remains at the end in case of N = 8.
(2) When cards are removed at the 1st round in case of N = 16, as shown in Fig. 2, eight cards remain.
The card in which 2 was written is removed next.
Answer the number written to the card which remains at the end in case of N = 16.
Moreover, answer the number written to the card which remains at the end, respectively in case of N = 32 and N = 64.
(3) When the cards in which 1, 3, and 5 are written at the 1st round are removed in case of N = 35, 32 cards remain.
The card in which 7 was written is removed next.
Answer the number written to the card which remains at the end in case of N = 35.
(4) When 36 cards are removed to the 1st round in case of N = 100, 64 cards remain.
Answer the number written to the card which remains at the end in case of N = 100.
(5) Answer the number written to the card which remains at the end in case of N = 2009.
Answer
(1) 8
(2) 16, 32, 64
(3) 6
(4) 72
(5) 1970
