For Geeks & Brainiacs #21- How Many Children -
A pizza delivery man named Mario (who also happens to be a mathematician) arrives at a house with a stack
of pizzas and rings the doorbell. Mr. White, the owner of the house, opens the door. Mario hears several
children inside playing together noisily. "Big family!" Mario says. "It's a birthday party. My children
are playing with friends from three other families." Mr. White replies. "My wife and I have the most
children, followed by the Browns, the Greens, then the Reds. Each family has a different number of
children." Mr. White adds.
"How many children are in there?" Mario asks. "Fewer than 18 and the product of the number of children
in each of the four families is the same as our house number." Mr. White answers with a grin. Mario looks
at the house number and does some calculations on his notepad. "That's not enough information. Is there
more than one child in the Red family?" Mario inquires. When Mr. White replies, Mario correctly states
the number of children that are there. So how many children are there?
HINT: Begin by making a list of all combinations of four different numbers where the sum
is less than 18.
HINT 2: Cross of combinations that produce a unique product when the four numbers are
multiplied together since the house number alone was not enough to solve the puzzle.
ANSWER:
14.
EXPLANATION: Each of the four families have a different number of children and there are
less than 18 children in all. The first step is to list all combinations of four unique numbers that
add up to less than 18 along with their product. If there was only one combination with the same product
as the house number, Mario would have known how many children there were right away. But since he required
additional information, there was more than one combination with the same product as the house number,
so all combinations with a unique product can be crossed off. Once Mario was told whether there was more
than one child in the Red family, he was able to solve the puzzle (i.e. only one combination remained).
Looking at the remaining combinations in the table below, the answer must have been yes and the only
combination possible is highlighted in yellow: 2 Reds, 3 Greens, 4 Browns, and 5 Whites for a total of 14
children in all.
| # Of Children (R's, G's, B's, W's) |
Product |
Group |
1, 2, 3, 4 |
24 |
Unique |
1, 2, 3, 5 |
30 |
Unique |
1, 2, 3, 6 |
36 |
Unique |
1, 2, 3, 7 |
42 |
Unique |
| 1, 2, 3, 8 |
48 |
Group A |
1, 2, 3, 9 |
54 |
Unique |
| 1, 2, 3, 10 |
60 |
Group B |
1, 2, 3, 11 |
66 |
Unique |
1, 2, 4, 5 |
40 |
Unique |
| 1, 2, 4, 6 |
48 |
Group A |
1, 2, 4, 7 |
56 |
Unique |
1, 2, 4, 8 |
64 |
Unique |
| 1, 2, 4, 9 |
72 |
Group C |
| 1, 2, 4, 10 |
80 |
Group D |
| 1, 2, 5, 6 |
60 |
Group B |
1, 2, 5, 7 |
70 |
Unique |
| 1, 2, 5, 8 |
80 |
Group D |
| 1, 2, 5, 9 |
90 |
Group F |
| 1, 2, 6, 7 |
84 |
Group E |
| 1, 2, 6, 8 |
96 |
Group G |
| 1, 3, 4, 5 |
60 |
Group B |
| 1, 3, 4, 6 |
72 |
Group C |
| 1, 3, 4, 7 |
84 |
Group E |
| 1, 3, 4, 8 |
96 |
Group G |
1, 3, 4, 9 |
108 |
Unique |
| 1, 3, 5, 6 |
90 |
Group F |
1, 3, 5, 7 |
105 |
Unique |
| 1, 3, 5, 8 |
120 |
Group H |
1, 3, 6, 7 |
126 |
Unique |
| 1, 4, 5, 6 |
120 |
Group H |
1, 4, 5, 7 |
140 |
Unique |
| 2, 3, 4, 5 |
120 |
Group H |
2, 3, 4, 6 |
144 |
Unique |
2, 3, 4, 7 |
168 |
Unique |
2, 3, 4, 8 |
192 |
Unique |
2, 3, 5, 6 |
180 |
Unique |
2, 3, 5, 7 |
210 |
Unique |
2, 4, 5, 6 |
240 |
Unique |
Do you have a
suggestion for this puzzle (e.g. something that should
be mentioned/clarified in the question or solution, bug, typo, etc.)?
