How many Children
"I heard some children playing in your backyard," said Jones. "Are they all yours?"
"Heavens no," exclaimed Smith. "My children are playing with friends from three other families in the neighborhood, although our family happens to be the largest. The Browns have a smaller number of children ... the Greens have a still smaller number ... and the Blacks family is the smallest of all."
"How many children are there all together?" asked Jones.
"Let me put it this way," said Smith. "There a fewer than eighteen children, and the product of the number of children in each of the four families
happens to be my house number ... which you saw when you arrived."
Jones began figuring on his notepad ... he then said, "I need more info ... is there more than one child in the Black family?"
As soon as Smith responded, Jones smiled and then stated the number of children in each family.
How many children were in each family?
- Zaux
"Heavens no," exclaimed Smith. "My children are playing with friends from three other families in the neighborhood, although our family happens to be the largest. The Browns have a smaller number of children ... the Greens have a still smaller number ... and the Blacks family is the smallest of all."
"How many children are there all together?" asked Jones.
"Let me put it this way," said Smith. "There a fewer than eighteen children, and the product of the number of children in each of the four families
happens to be my house number ... which you saw when you arrived."
Jones began figuring on his notepad ... he then said, "I need more info ... is there more than one child in the Black family?"
As soon as Smith responded, Jones smiled and then stated the number of children in each family.
How many children were in each family?
- Zaux
Labels: logic, SharedPuzzle
posted by ToM at
4:23 PM
15 Comments:
If Smith said Yes:
2,3,4,5 for 14 kids
If Smith Said No:
1,4,5,6 for 16 kids
(Smith's house # is 120)
Details to follow.....
Cam
This post has been removed by the author.
How many Children
Information provided:
-4 families (a,b,c,d)
d > c > b > a
18 > a+b+c+d
a*b*c*d = P
a,b,c,d are unknown
Product,P, (the house number) is known
Analysis:
smallest possible sum is
1+2+3+4 = 10
since given the P Jones may answer the question provide he knows if a is 1 or >1 then there must be at least 2 solutions where
a*b*c*d=P, at least one where a =1 and at least one where a > 1
minimum sum where a > 1
2+3+4+5= 14
identify max a
minimum sum with largest a is
a+ (a+1) + (a+2) + (a+3)= 4a+6
18 > 4a+6
12> 4a
3> a
a must be 1 or 2
Smallest possible product for
a*b*c*d when a>1 is
2*3*4*5=120
Check for range of products when a =1. Sums of 17 will make largest products
1,2,3,11 P=77
1,2,4,10 P=80
1,2,5,9 P=90
1,2,6,8 P=96
1,3,4,9 P=108
1,3,5,8 P=120
1,3,6,7 P=126
1,4,5,7 P=140
140 is largest P found for a=1
Thus we need to only look at combos for a=2 where 140 >= P
2,3,4,5 =120
2,3,4,6=144 , stop here since all other products with a=2 must be larger
So product must be 120.
Factors of 120 are
1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120
largest possible d is 11, thus we need only consider factors below this
10,4,3,1 S=18 invalid
8,5,3,1 S=17 OK
6,5,4,1 S=16 OK
This leaves 3 possibilities where P=120:
1,3,5,8 S=17
1,4,5,6 S=16
2,3,4,5 S=14
once he is provided with a the answer is known, however if a =1 there would be two possibilities (more info would still be required), thus a must be 2
Leaving the only possible combo: 2,3,4,5
Answer:
Blacks have 2 kids
Greens have 3 kids
Browns have 4 kids
Smith has 5 kids.
Hopefully no mistakes this time .My previous post was in error,(missed the 1,3,5,8 combo). The solution is, in fact, unique.
Cam
I withdrew my comment after seeing that I'd also made the same mistake as Cam originally did, and as Cam has made a more thorough analysis.
Chris,
Strangely enough, we both missed the 1,3,5,8 combo.
As a result, you may want to reconsider this statement:
"It is also obvious
that there was only going to be one common product" ;)
Cam
Hi Cam, LOL - you got me there! In fact I only thought it obvious with hindsight - which "obviously" isn't the 20-20 that the saying suggests. Alternative excuse is that it's very late (nearly 6 am) and I'm stoned.
Cam ... nice work ... you ever considered applying to Mensa?... heh heh!
Chris ... question ... how do you remove a post in Trick of Mind?
Hi Zaux. You need to sign in with a Googlemail account. They're free. Then you can the post and re-edit your own blog, delete anyone's comment on it (optionally completely - i.e. no "this post has been removed by the author.") and delete your own comments on someone elses blog, but leaving a "this post has been removed by the author" tell tale.
You also can access a decent interface to peruse all the blogs. I blog per line, 25 blogs per page.
You can also post pictures in the header of you own blog (Ragknot does this sometimes), but I've forgotten how to do that.
Hi Zaux. I agree that Cam is very good at solving maths problems and at presenting his reasoning. He is also very good at not making mistakes (that's my downfall - I'm over-confident, and haven't come to terms with my 57 year old brain not being as good as it once was). I'm confident that Cam easily surpasses Mensa's IQ requirement. Mensa are for the top 2%, I guess that Cam is in at least the top 0.5%).
... and thanks for the problem Zaux. It was a proper puzzle and more than worthy of the space taken.
Aww shucks... now you guys have me blushing.:)
The answer to, if I have ever considered applying to Mensa is:
Not seriously. The concept of the organization doesn't really appeal to me. No offense to any Mensa members out there, but it sounds a tad bit too elitist to me (most clubs focus on an activity, and let you join even if you stink at it), plus I can't imagine that they throw very good parties. Of course, I could be completely wrong about this.
Cam
Cam ... I agree ... I wouldn't want to belong to Mensa... it was simply a compliment. Are you a math teacher or someone who just loves math?
...
Zaux,
I am not a math teacher. I just enjoy math and challenging puzzles.
Kudos to you for posting some great puzzles.
Cam
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