Ages And Cakes

Logic Level 1

Mr and Mrs Tan have five children born in consecutive years (i.e. of consecutive ages). One day, there was a piece of cake for Mrs Tan but one of their five children ate it. This is what we know:

Eric and the culprit are of one year apart.
Charles and the culprit are three years apart.
Alfred is neither the oldest nor the youngest.
Darius is older than the culprit.
Brenda is the middle child.

Who stole the cake?

Alfred Brenda Charles Darius Eric

2 solutions

Noel Lo
Jun 2, 2016

Simple method:

Since Eric, Charles and Darius are being compared with the culprit, none of the three is the culprit, leaving only Alfred and Brenda. But if Brenda is the culprit, then looking at the statement about Charles, we have an impossible scenario. Since Brenda is the middle child, she is the third child of five. This means she can only be at most two years older or two years younger than any of her brothers since all five are of consecutive ages. She and Charles cannot possibly be three years apart. Hence Alfred must be the culprit.

More complicated method:

Since Brenda is the middle child, she is the third child of five. Since Alfred is neither the oldest nor the youngest, he is either the second or fourth child. Suppose Alfred is the second child. Based on the statement about Charles, we have (Charles, culprit) = (1st, 4th), (4th, 1st), (5th, 2nd) as possible solutions. If (Charles, culprit) = (1st, 4th), then Charles and Darius would be fighting for the oldest position as Darius is older than the fourth child but Alfred is 2nd and Brenda is 3rd. If (Charles, culprit) = (4th, 1st), then Eric would be fighting with Alfred for the second oldest position as Eric and the culprit are of consecutive ages. If (Charles, culprit) = (5th, 2nd), then Eric would be fighting with Darius for the oldest position.

Hence it is impossible for Alfred to be the second child. He must be the fourth child. Hence (Charles, culprit) = (1st, 4th), (2nd, 5th), (5th, 2nd). If (Charles, culprit) = (2nd, 5th), Eric and Alfred would be fighting for the fourth position while if (Charles, culprit) = (5th, 2nd), Eric and Darius would be fighting for the oldest position. Hence Charles is the oldest while the culprit is the fourth child who is Alfred himself. This means Darius is the second child while Eric is the youngest. This is the only workable solution:

Charles (oldest), Darius, Brenda (middle), Alfred , Eric (youngest)

I like that you gave two solutions. I think they could have been both told even more synthetically but anyway it's well enough how you stated them anyway. It's interesting to compare what was important in both solution to deduce everything. This leads to an understanding of deduction in general that is to "meta-logical" understanding anyway and cute solutions.

A A - 5 years ago

Stewart Gordon
Jun 10, 2016

We are told that Eric, Charles and Darius are different ages from the culprit. Brenda is the middle child, so can't be three years apart from Charles. So it's Alfred. Easy!

Full order of them

Alfred is neither the middle child, the oldest nor the youngest, so he is either the second oldest or the second youngest. If he were the second oldest, then this would force both Eric and Darius to be the oldest, which is impossible. Hence Alfred is the second youngest. It immediately follows that Eric is the youngest, Charles is the oldest, and Darius is the second oldest.

From youngest to oldest: Eric, Alfred, Brenda, Darius, Charles.

Ages And Cakes

2 solutions

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