I've chosen a two-digit positive integer n (i.e. 10 - 99) and tell its first digit to Ojas, its second digit to Paddy, and its digit sum to Quentin. The following conversation occurs, in which all three of them have perfect deduction skills, and "only now" means since the previous piece of knowledge:

Paddy: "The number isn't prime."
Ojas: "It isn't a perfect square."
Paddy: "I'm not sure if it is a triangle number or not."
Quentin: "I know what it is. But only now."
Ojas: "I only now know what it is."

What is n?

Paddy: "The number isn't prime."

This rule out all numbers with units-digit 1,3,7,9 

Ojas: "It isn't a perfect square." 

This rule out all numbers with tens-digits 1, 2, 3, 6. It doesn't rule out 4 and 8 because the squares 49 and 81 were already out by Paddy's assertion

Paddy: "I'm not sure if it is a triangle number or not." 

On the remaining numbers only 45, 55, 78 are triangular numbers. If Paddy is not sure if the number is triangular it implies that the units-digit number is 5 or 8, but he can't exclude not-triangular numbers finishing in 5 or 8.

Quentin: "I know what it is. But only now."

With Paddy's second assertion Quentin has came to know that the number is finishing in 5 or 8. So the possible numbers are:

45, 48, 55, 58, 75, 78, 85, 88, 95, 98. 

Quentin knows the sum of both digits. 

If the chosen number would have been 98, 88 he would have know the number before Paddy's second assertion, as the sum of digits is unique for the set of numbers then remaining. 

If the chosen number would have been 48, 75 (either one or the other) and 85 or 58 (one or the other) Quentin would not have been able to know the number as both candidates in each group have the same sum and fit within the other conditions. 

If he has been able to elaborate the number only after 2nd Paddy's assertion is because that leads to a unique candidate for the sum he knows. This happen with numbers 45, 55, 78, 95, with respective sums of 9, 10, 15, 14. As Quentin knows the sum he knows the number. 

Ojas: "I only now know what it is."

Ojas has follow the conversation so he knows that the candidate is one of these four numbers (45, 55, 78, 95) 

Before Quentin's assertion he could not elaborate the numbers as the candidates were (45, 48, 55, 58, 75, 78, 85, 88, 95, 98) and he did know only the tens-digit. But there were two candidates for each possible tens-digit. 

But Quentin's assertion has ruled out six of these numbers leaving only in field 45, 55, 78, 95. As Ojas knows the tens-digit he is now able (as Quentin) to elaborate the number. 

The problem here is we are not. 

We do not know the sum or the tens-digit so we are left with four candidates. 

Is the enunciate offering us something more? Here is my bizarre answer. Perhaps. It is giving the names of the people speaking. If we assign a value to each character (a=1, b=2, c=3 ...) we get something weird: 

Paddy=50

Quentin=99

Ojas=45

Quentin has finished the task in first place and Ojas has arrived to the number. Paddy, as ourselves, has been left in the middle of the way. Now we, as Paddy, also get an answer: 

n=45

Edited on March 14, 2019, 11:40 am

Posted by armando on 2019-03-14 07:54:37