Can you fill in the blanks in the following sequence?

1,1,4,2,7,3,17,4,20,?,37,6,47,5,51,?,61,3,70,3,9?,2,100,1

(In reply to

Answer by K Sengupta)

At the outset, we divide the given numbers in the sequence in twelve pairs as follows:

(1,1), (4,2), (7,3), (17,4), (20,?), (37,6), (47,5), (51,?), (61,3), (70,3), (9?,2), (100,1)

We leave out the pair (9?,2) for now and observe that the second element of each pair is obtained by writing the number of letters in the Roman numeral representation of the first element. For example, 17 is written as XVII in Roman representation and this representation has 4 letters which gives the second element.

We now observe that the roman representation of 20 is XX, which contains two letters and therefore the 5th pair is (20, 2). Similarly, the roman representation of 51 is LI, which contains two letters and therefore the 8th pair is (51, 2).

We will now consider the pair (9?, 2) and writing the numbers 90 to 99, we will note the letters inclusive of each of the said representations in terms of the following table:

Number Roman Numeral No. Of letters

Representation

90 XC 2

91 XCI 3

92 XCII 4

93 XCIII 5

94 XCIV 4

95 XCV 3

96 XCVI 4

97 XCVII 5

98 XCVIII 6

99 XCIX 4

From the above table, the only number whose Roman representation contains two letters is 90, and thus the 11th pair is (90, 2) and the digit corresponding to the question mark is 0.

Consequently, the respective numbers/digits that correspond to the marks of interrogation are 20, 51 and 0.

*Edited on ***May 10, 2008, 4:03 pm**