The following conversation between Bob and Jack in a Chinese restaurant was overheard:

logic/lunch.specials New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

The following conversation between Bob and Jack in a Chinese restaurant
was overheard:

Bob: When did you and Cindy first meet?
Jack: In college, when I was a senior and she was a freshman.
Bob: How many kids do you and Cindy have?
Jack: Same as what you have.
Bob: And how old are they?
Jack: I'll let you figure this one out.  The sum of their ages is equal
      to the number of lunch specials on the menu, and the product of
      their ages is equal to my age.
Bob: How old are you?
Jack: (A brief pause) Actually, knowing my age wouldn't help you much.
      However, this might help: we have no twins.

Amazingly, the above conversation allows you to answer Bob's questions.
So here are the questions for you:
(1) How many kids did Jack and Cindy have?
(2) How many lunch specials were on the menu?
(3) How old were the kids?
(4) How old was Jack? [Editor: numbered (3) in the original]
Remark: all ages are in integers.

-- Robert A. Crawford <globalking@resnet.gatech.edu>

Show solution

With two children knowing the sum and product of the ages will always
be enough, so that there must be at least three children.  Furthermore,
if the sum and product of the ages are known, there are at least two
possible answers, only one of which consists of all different ages.

The first number for which two factorizations with the same sum exist,
only one of which consisting of three different numbers is 40.  The
factorizations are 1, 5, 8 and 2, 2, 10 - both having the sum 14.

14 can be broken into sums of three integers only as follows:

1 1 12 - product  12
1 2 11 - product  22
1 3 10 - product  30
1 4  9 - product  36
1 5  8 - product  40
1 6  7 - product  42
2 2 10 - product  40
2 3  9 - product  54
2 4  8 - product  64
2 5  7 - product  70
2 6  6 - product  72
3 3  8 - product  72
3 4  7 - product  84
3 5  6 - product  90
4 4  6 - product  96
4 5  5 - product 100

From the fact that knowing Jack's age is not enough and that there are
no twins Bob can deduce Jack's age to be 40 and the children's ages to
be 1, 5 and 8.

Therefore:
1. Jack and Cindy have 3 children.
2. There are 14 lunch specials.
3. The children are 1, 5 and 8 years old.
4. Jack is 40.

-- Ilan Mayer <ilan@isgtec.com>

There is only one answer with reasonable ages.  By reasonable, I mean
considering human lifespans and possible child-bearing ages.  From the
fact that Jack and Cindy were in College together, I'm also that Jack
and Cindy were reasonably close in age.  I'm ruling out the possibility
that Jack was 90 and Cindy was say, 35.  (There are 2 alternative sets
of answer that work with Jack being 90 and there is one set of answers
where Jack would be 12.  Presumable there are solutions for larger and
larger ages.)

The questions and answers imply that the number of children, the sum of
their ages, and the product of their ages is not enough to uniquely
identify them.  But given the additional information that there are no
twins (no two children with the same age), we can determine their ages.
We can rule out 2 children because the sum and product would always
uniquely identify the ages.  We can rule out 5 or more because no twins
===> product of ages >= 1*2*3*4*5 = 120 = Jack's age.

There's not much else you can do besides try out the possibilities for
3 and 4 children.  It turns out that the only reasonable answer is that
there are 3 children with ages 1, 5, 8.  There is one other set of three
numbers that has the same sum and product, namely 2, 2, 10.  So knowing
the number of children, the product of their ages, and their sum is not
enough to allow Bob to determine their ages.  But the additional
information that there are no twins, rules out 2, 2, and 10 meaning
that their ages must be 1, 5, and 8.

(1) How many kids did Jack and Cindy have?
Three

(2) How many lunch specials were on the menu?
Fourteen

(3) How old were the kids?
1, 5, and 8

(4) How old was Jack?
Forty.

-- Glenn Rhoads <rhoads@john.rutgers.edu>  (October 1998)