A riddle about three children

Question

We have three children: $A$ , $B$ , and $C$.

Child $A$ says: "I ate more sweets than $B$ , and $B$ ate more sweets than $C$."

Child $B$ says: "$C$ ate more sweets than me , and $C$ also ate more sweets than $A$."

Child $C$ says: "$B$ ate more sweets than me , also $A$ and I ate the same amount of sweets."

Suppose that between the two children who ate less sweets say more right.

Which of the following sentences can not be correct?

1- $C$ ate the most amount of sweets.

2- $B$ ate the most amount of sweets.

3- $A$ ate more sweets than $B$.

4- $B$ ate more sweets than $A$.

5- $B$ ate more sweets than $C$.

6- $C$ ate more sweets than $B$.

7- $C$ ate more sweets than $A$.

8- $A$ ate more sweets than $C$.

9- All children ate an equal amount of sweets.

Answer 1

Child A says: " I ate sweets more than B , and B ate sweets more than C "

Child B says: " C ate sweets more than me , also C ate sweets more than A "

Child C says: " B ate sweets more than me , also I and A ate sweets equal amount "

Suppose between the two children who ate less sweets say more right.

Two children who ate less sweets ?

A: ate more than B B: Ate more than C

B: c ate more than B

C: ate more than A

B: ate more than C

C: ate same as A

Which children ate less sweets ?

Here the number represents how many times the letter ate less

‘A’: 3 ‘B’: 3 ‘C’: 3

As you can see, All letters ate equally less sweets.

Therefore, the answer is 9 - All children ate sweets equal amount.

Answer 2

What each child said:

• Child A. A > B and B > C. A ate the most
• Child B. C > B and C > A. C ate the most
• Child C. B > C and A = C. B ate the most

There are rules that can be inferred from the statements. Any of the following sentences that breaks one of these rules cannot be correct:

(1) These 2 children must say at least 1 thing right (1 or 2 right things). This comes from “Suppose between the two children who ate less sweets say more right”.

(2) At most one child says 2 right things. This is the case because every child says at least one statement that contradicts at least one statement from every other child

(3) The child who eats the most cannot say anything right. Whilst “Suppose between the two children who ate less sweets say more right” implies that the child who eats the most could say one thing right, this would mean that the other two children must say two right things, which is impossible.

Going through the 9 sentences:

1- C ate sweets more than A and B. C > A and C > B Child A: says either 0 or 1 right things. B > C is wrong, but we don’t know if A > B Child B: says 2 things right Child C: says 2 wrong things If A > B, then the rules are followed

2- B ate sweets more than A and C. B > A and B > C Child A: says 1 thing right (B > C) Child B: says 0 or 1 things right. C > B is wrong. We don’t know if C > A Child C: says 1 or 2 things right. B > C is right. We don’t know if A = C If A ≥ C, then Child B says 0 things right, and other children say at least 1 thing right. So the rules are followed.

3- A ate sweets more than B. A > B Child A: says at least one thing right. To avoid contradicting rule (3), then C > A > B. This makes Child C the child who ate the most and have 2 incorrect statements, whilst Childs A and B have one correct statement. The rules are followed

4- B ate sweets more than A. This can follow the same pattern as in sentence 2 to follow the rules.

5- B ate sweets more than C. This can follow the same pattern as in sentence 2 to follow the rules.

6- C ate sweets more than B. This can follow the same pattern as in sentence 1 to follow the rules.

7- C ate sweets more than A. This can follow the same pattern as in sentence 1 to follow the rules.

8- A ate sweets more than C. If A > C, and also B > A > C, then this can follow one of the patterns mentioned in sentence 2 to follow the rules.

9- All children ate sweets equal amount. A = B = C In this case, both child A and B said 0 correct things. This contradicts rule (1) and hence must always be false.

Hence, by going through all the available options, we can conclude the only sentence that cannot ever be correct is sentence 9.