The rows of letters can be put in order using a Lottery “Amidakuji”.
For example, the row ABCDE of letters is replaced along with the row of letters ECDBA with the Lottery of Fig. 1.
The thick line in the figure is a route showing the way of A.
(1) Connect two Lotteries of Fig. 1 lengthwise and make a Lottery as shown in Fig. 2.
How is the row of letters ABCDE located in a line is replaced with this Lottery?
Answer the new row of letters.
(2) With the Lottery which connected some Lotteries of Fig. 1 lengthwise, when the row of letters ABCDE are rearranged, it became the same row of letters as original ABCDE.
How many Lotteries of Fig. 1 are connected lengthwise in this case?
Answer the least number.
(3) There is a Lottery which replaces the row of letters ABCDEFGHIJ along with the row of letters BCAJFGHIED.
With the Lottery which connected this Lottery lengthwise, when the row of letters ABCDEFGHIJ are rearranged, it became the same row of letters as original ABCDEFGHIJ.
How many Lotteries of this Lottery are connected lengthwise in this case?
Answer the least number.
For example, the row ABCDE of letters is replaced along with the row of letters ECDBA with the Lottery of Fig. 1.
The thick line in the figure is a route showing the way of A.
Answer the following questions.
(1) Connect two Lotteries of Fig. 1 lengthwise and make a Lottery as shown in Fig. 2.
How is the row of letters ABCDE located in a line is replaced with this Lottery?
Answer the new row of letters.
How many Lotteries of Fig. 1 are connected lengthwise in this case?
Answer the least number.
(3) There is a Lottery which replaces the row of letters ABCDEFGHIJ along with the row of letters BCAJFGHIED.
With the Lottery which connected this Lottery lengthwise, when the row of letters ABCDEFGHIJ are rearranged, it became the same row of letters as original ABCDEFGHIJ.
How many Lotteries of this Lottery are connected lengthwise in this case?
Answer the least number.
Answer
Solution
(1) A table is used to arrange the problem in order.
The row of letters ABCDE is replaced along with the row of letters ECDBA with the Lottery of Fig. 1.
As shown in a table below, it is that each letter in the row of letters ABCDE is replaced with each of ECDBA, respectively by one operation.
Connecting two Lotteries of Fig. 1 means that the same operation is repeated twice.
As shown in a table, E is changed to A, C to D, D to B, B to C, and A to E, respectively.
Therefore, ABCDE is replaced with ADBCE.
(2) As shown in a table, operation is repeated following (1).
When a letter returns to the original position, the operation of the letter is stopped.
For example, since A and E have returned to the original position by the 2nd time, operations are stopped.
As you can see the table the result of operation is filled out, there are two kinds of letter, one is the letter which returns to the original position by 2 times, and another is the letter which
returns by 3 times.
Therefore, it is a time of operating by the number of times of the common multiple of 2 and 3 that all letters return to the original position at the same time.
Since the least number of times is the least common multiple which is six, there are six pieces of lotteries to be connected.
(3) Operation is being continued like (2), the result is shown as in a table.
The number of times that each character returns to the original position will be three kinds, 2 times, 3 times, and 5 times.
Therefore, since the minimum number of times which all letters return to the original position simultaneously turns into the least common multiple of 2, 3, and 5, which is 30, the number of lotteries connected is 30 pieces.
(1) ADBCE
(2) Six pieces
(3) 30 pieces
(2) Six pieces
(3) 30 pieces
(1) A table is used to arrange the problem in order.
The row of letters ABCDE is replaced along with the row of letters ECDBA with the Lottery of Fig. 1.
As shown in a table below, it is that each letter in the row of letters ABCDE is replaced with each of ECDBA, respectively by one operation.
Connecting two Lotteries of Fig. 1 means that the same operation is repeated twice.
As shown in a table, E is changed to A, C to D, D to B, B to C, and A to E, respectively.
Therefore, ABCDE is replaced with ADBCE.
(2) As shown in a table, operation is repeated following (1).
When a letter returns to the original position, the operation of the letter is stopped.
For example, since A and E have returned to the original position by the 2nd time, operations are stopped.
As you can see the table the result of operation is filled out, there are two kinds of letter, one is the letter which returns to the original position by 2 times, and another is the letter which
returns by 3 times.
Therefore, it is a time of operating by the number of times of the common multiple of 2 and 3 that all letters return to the original position at the same time.
Since the least number of times is the least common multiple which is six, there are six pieces of lotteries to be connected.
(3) Operation is being continued like (2), the result is shown as in a table.
The number of times that each character returns to the original position will be three kinds, 2 times, 3 times, and 5 times.
Therefore, since the minimum number of times which all letters return to the original position simultaneously turns into the least common multiple of 2, 3, and 5, which is 30, the number of lotteries connected is 30 pieces.
