How old is Ann?
Dean Messing
writes:
>In article <benny.6...@dash.mitre.org> be...@bartok.mitre.org
(Robert Benedetto) writes:
> >Meir Feder <AER...@TECHNION.BITNET> writes:
> > >
> > > Mary`s and Ann`s ages compined total 44 years. Mary is twice
> > > as old as Ann was when Mary was half as old as Ann will be
> > > when Ann is three times as old as Mary was when Mary was three
> > > times as old as Ann. How old Ann is?
> > >
> > > Meir Feder, P.O.Box 3456, Haifa, Israel.
> >
> > Ann is 9 and 3/7 yrs old (=66/7)
>
> I had Ann at 16.5. When Mary was three times as old as Ann, Mary was
> 16.5 and Ann was 5.5. Ann will be 49.5 when she's three times as old
> as that. When Mary was 24.75 (half of 49.5), Ann was 13.75, which is
> indeed half of Mary's current age of 27.5.
I have two qualms with the original statement of the problem.
First, a past-tense clause should have been future-tense if, as
I believe, Ann's age is indeed 9 3/7 years old.
Second, to whom does the final "Ann" in the problem statement refer?
I.e., to whom does the Ann in "... three times as old as Ann." refer?
I claim it must be the current Ann - the one whose age we wish to determine.
Mr. Mayville assigns this Ann an age of 5.5 yrs. which seems to indicate that
he treated her as an additional unknown (in addition to the 3 Anns and 3 Marys)
that the problem encompasses.
What I mean will be made clear by the following analysis.
We can view the problem as giving an age relationship between Ann and Mary
at 3 different times in their lives. Let A0 and M0 refer to the ages of the
current Ann and Mary. Thus we want to find A0.
Next there are the Ann and Mary who live contemporaneously when Mary "was half
as old as Ann will be ..." Call their ages A2 and M2.
Finally there is the Ann who "is three times as old as Mary
when Mary was three times as old as Ann IS NOW".
^^^^^^
Assign her age to be A1. Also there is a Mary in this sentence whose
age we will call M1. Note: they are NOT contemporaneous.
Notice that I have added the words "IS NOW" to identify
the final Ann as A0. Otherwise we have yet another variable A3 and the
number of equations which the problem defines is one fewer than the number
of variables, making the problem indeterminate.
The final Ann could not be A1 else we would have "A1 is three times
as old as M1 when M1 is three times as old as A1" which makes A1 = M1 = 0,
which is nonsense.
It is conceivable that the final Ann is A2. I will handle that case later.
For now I assume the final Ann is A0.
Lets restate the problem using my variable names, the tense
changes I claim are appropriate, and a bit of re-formatting.
M0 and A0 sum to 44.
M0 is twice as old as A2 WILL BE when (**)
M2 is half as old as A1 will be when
A1 is three times as old as M1 was when
M1 is three times as old as A0.
How old is A0?
I have capitalized the tense change.
The "when" marked (**) indicates that A2 and M2 are contemporaneous.
The other "when's" also indicate contemporaneity ... between A1 and A1,
and between M1 and M1 ... which are trivially true.
The first equation is clearly A0 + M0 = 44. (1)
Each line defines yet another independent eqn. M0 = 2*A2. (2)
M2 = A1/2. (3)
A1 = 3*M1. (4)
M1 = 3*A0. (5)
Unfortunately there seems to be no connection between
A2 and M2. But there is! They are contemporaneous.
Seeing this is the solution to the puzzle. Hence M2-A2 = M0-A0 (6)
since the difference between Ann and Mary's age is
constant.
We now have 6 eqns in 6 unknowns.
A sketch of their solution is:
A1 = 9*A0 (a) (from eqs. 4 and 5).
3*M0 = 2*M2 + 2*A0 (b) (eliminate A2 from 2 and 6 and rearrange).
3*M0 = A1 + 2*A0 (c) (eliminate M2 from b and 3).
3*M0 = 11*A0 (d) (plug a into c).
3*(44-A0) = 11*A0 (e) (eliminate M0 form 1 and d)
==> A0 = 9 3/7 yrs.
==> M0 = 34 4/7 yrs.
Hence A1 = 9*A0 = 84 6/7 yrs.
M1 = 3*A0 = 28 2/7 yrs.
A2 = 17 2/7 yrs.
M2 = 42 3/7 yrs.
It is now clear that the past tense clause referring to A2 in the original
should have been future tense. Mary(M0) is as old as Ann(A2) WILL BE .... &c.
since A2 is older than A0. The original correctly got A1 older than A0.
and correctly got M1 younger than M0.
The reason I criticize this is because I originally tried to solve
the puzzle with age inequalities based on the given tenses.
The attempt yielded the inconsistency "Ann was both older than 22 and younger
than 10" which caused me to think there was an error in the puzzle and give up.
Finally, if we assume the "final Ann" is A2 instead of A0
then eqn. (5) above becomes M1 = 3*A2. (5a)
Now the "solution" is:
3*M0 = 2*M2 + 2*A0 (a) (eliminate A2 from 2 and 6 and rearrange).
3*M0 = A1 + 2*A0 (b) (eliminate M2 from a and 3).
3*M0 = 2*M1 (c) (eliminate A2 from 2 and 5a and rearrange).
2*A1 = 4*M1 - 4*A0 (d) (c --> b , mply x2 and rearrange)
2*A1 = 6*M1 (e) (mply (4) x2)
==> 0 = 2*M1 + 4*A0
==> one of M1 and A0 is negative ... nonsense.
Hence the "final Ann" must be A0.
Dean Messing
Digital Signal Processing Research
TekLabs, Tektronix Inc.
E-Mail: de...@medulla.labs.tek.com
Kevin James Mayville
>> > > as old as Ann was when Mary was half as old as Ann will be
>> > > when Ann is three times as old as Mary was when Mary was three
>> > > times as old as Ann. How old Ann is?
1) M0 + A0 = 44
2) M5 = 3*A5
Now, "Mary is" give you M0..."Twice as old as Ann was" make it
M0 = 2*Ai (some intermediate age)
"When Mary was" = Mi (simultaneous with Ai) "half as old as Ann will be when
Ann is three times as old as Mary was when Mary was..." gives you
2*Mi = 3*M5 = 9*A5
Mi = 4.5*A5
Since the difference between their ages is constant, we have Mi - Ai = 2*A5
(from Eqn. 2). So, Ai = 2.5*A5.
Substituting, we get M0 = 5*A5
Since M0 - A0 = 2*A5, A0 = M0 - 2*A5, or 3*A5. We're now down to:
8*A5 = 44, or A5 = 5.5. Since A0 = 3*A5, A0 = 16.5
>I have two qualms with the original statement of the problem.
>First, a past-tense clause should have been future-tense if, as
>I believe, Ann's age is indeed 9 3/7 years old.
>Second, to whom does the final "Ann" in the problem statement refer?
>I.e., to whom does the Ann in "... three times as old as Ann." refer?
>I claim it must be the current Ann - the one whose age we wish to determine.
Your answer is already seriously in doubt if it requires that the problem
was stated incorrectly, especially since my answer fits the problem as stated.
>We can view the problem as giving an age relationship between Ann and Mary
>at 3 different times in their lives. Let A0 and M0 refer to the ages of the
>current Ann and Mary. Thus we want to find A0.
There are only 2 points in their lives that are "simultaneous" stated in the
problem. A0 + M0 = 44, and M5 = 3*A5. Where do you get a third one from?
>Next there are the Ann and Mary who live contemporaneously when Mary "was half
>as old as Ann will be ..." Call their ages A2 and M2.
>
>Finally there is the Ann who "is three times as old as Mary
>when Mary was three times as old as Ann IS NOW".
Whoops!! Adding words to the problem. Tut-tut. "when Mary was three times as
old as Ann" just states that at some point Mary was 3 times older. Has nothing
to do with whether it's NOW, in fact it implies that it isn't.
>Assign her age to be A1. Also there is a Mary in this sentence whose
>age we will call M1. Note: they are NOT contemporaneous.
>
>Notice that I have added the words "IS NOW" to identify
>the final Ann as A0.
However, there is nothing in the problem to indicate that the final Ann
is A0, and a fair bit to indicate that she isn't. You've formed your
equations incorrectly.
Col. G. L. Sicherman
] > > >
] > > > Mary`s and Ann`s ages compined total 44 years. Mary is twice
] > > > as old as Ann was when Mary was half as old as Ann will be
] > > > when Ann is three times as old as Mary was when Mary was three
] > > > times as old as Ann. How old Ann is?
] > > >
] > > > Meir Feder, P.O.Box 3456, Haifa, Israel.
> > 16.5 and Ann was 5.5. Ann will be 49.5 when she's three times as old
> > as that. When Mary was 24.75 (half of 49.5), Ann was 13.75, which is
> > indeed half of Mary's current age of 27.5.
>
> I have two qualms with the original statement of the problem.
> First, a past-tense clause should have been future-tense if, as
> I believe, Ann's age is indeed 9 3/7 years old.
> Second, to whom does the final "Ann" in the problem statement refer?
> I.e., to whom does the Ann in "... three times as old as Ann." refer?
> I claim it must be the current Ann - the one whose age we wish to determine.
No, in everyday usage "when Mary was three times as old as Ann" means
at a time when Mary was three times as old as Ann was. This is also
what it normally means in puzzles. So the previous answer is correct,
and so are the tenses in the statement of the problem.
By the way, this puzzle was composed by Sam Loyd as a sequel to his famous
"How old is Ann?" puzzle. It's called "How old is Mary?", which is how it
originally ended. Give credit where it's due!
Now here's another one by Sam Loyd:
They were in a magnificently decorated room in the West End
of London. They approached each other from opposite directions.
Presently they met, and careless of the fact that dozens of
eyes were watching them, they kissed each other with a resound-
ing smack!
The meeting seemed to bring them perfect peace; but alas,
alack! they had scarcely been side by side above twenty seconds
when a man approached with the fire of battle in his eye. With
cool insolence he raised the stick he carried, and then--oh,
horror!--he struck a sharp, quick blow, and the pale one was
sent spinning several feet away.
The other neither screamed nor fainted. There was no heart-
breaking, no resentment; not even a murmur was heard, because---?
--
Col. G. L. Sicherman
g...@hrmso.att.COM
Jonathan M Lennox
>
>Now here's another one by Sam Loyd:
>
> They were in a magnificently decorated room in the West End
> of London. They approached each other from opposite directions.
> Presently they met, and careless of the fact that dozens of
> eyes were watching them, they kissed each other with a resound-
> ing smack!
> The meeting seemed to bring them perfect peace; but alas,
> alack! they had scarcely been side by side above twenty seconds
> when a man approached with the fire of battle in his eye. With
> cool insolence he raised the stick he carried, and then--oh,
> horror!--he struck a sharp, quick blow, and the pale one was
> sent spinning several feet away.
> The other neither screamed nor fainted. There was no heart-
> breaking, no resentment; not even a murmur was heard, because---?
>
Jonathan Lennox
jm...@cunixa.cc.columbia.edu
Michael Neylon
(Col. G. L. Sicherman) writes...
>Now here's another one by Sam Loyd:
>
> They were in a magnificently decorated room in the West End
> of London. They approached each other from opposite directions.
> Presently they met, and careless of the fact that dozens of
> eyes were watching them, they kissed each other with a resound-
> ing smack!
> The meeting seemed to bring them perfect peace; but alas,
> alack! they had scarcely been side by side above twenty seconds
> when a man approached with the fire of battle in his eye. With
> cool insolence he raised the stick he carried, and then--oh,
> horror!--he struck a sharp, quick blow, and the pale one was
> sent spinning several feet away.
> The other neither screamed nor fainted. There was no heart-
> breaking, no resentment; not even a murmur was heard, because---?
Hint: Momentum and physics problems abound in this situation....
Solution: It's in the middle of a pool game (either snookers or eight ball, but
since its in London, i would guess the former). Two billard balls have just
hit each other, and the man (with his cue stick), knocks the cue ball ("pale")
away...
+--------====---======-=====----=====---====-====-----+------------------------+
|Mike / -- ) / -- / / __/ / __/ / _ _ / |'Man was never deemed to|
| / // / / // / / / / / __ / // // / | know calculus... |
| / -- / / -- / / /__ / /_/ / / // // / | and infinite series are|
| / / / / / / / / / // // / Neylon | proof enough...' |
+---======-======-=====----=======-===-==-==----------+------------------------+
Stephen H. Landrum
people state their ages in whole years, no matter how many months
they are into their current year.
My solution follows:
For my solution, I get Mary at 28 years, and Ann at 16. Mary is
more than 12 but less than 13 years older than Ann.
> Mary`s[28] and Ann`s[26] ages compined total 44 years. Mary[28] is twice
> as old as Ann[14] was when Mary[27] was half as old as Ann[54] will be
> when Ann[54] is three times as old as Mary[18] was when Mary[18] was three
> times as old as Ann[6]. How old Ann is?
This is a much more satisfying answer and I think more along the
lines of what I would have expected from a Sam Lloyd puzzle.
--
Stephen H. Landrum VOICE: (415) 813-8909
Domain: slan...@ntg.com UUCP: ...apple!ntg!slandrum
USNAIL: New Technologies Group Inc. 2468 Embarcardero Way, Palo Alto CA 94303
Meir Feder
>] > > > as old as Ann was when Mary was half as old as Ann will be
>] > > > when Ann is three times as old as Mary was when Mary was three
>] > > > times as old as Ann. How old Ann is?
>] > > >
>] > > > Meir Feder, P.O.Box 3456, Haifa, Israel.
>] > >
>By the way, this puzzle was composed by Sam Loyd as a sequel to his famous
>"How old is Ann?" puzzle. It's called "How old is Mary?", which is how it
>originally ended. Give credit where it's due!
>
T H A N K S A L O T
I was waiting a log time for this kind of reply and I am very glad
to give the credit to Sam Loyd.
Kevin James Mayville
>more than 12 but less than 13 years older than Ann.
>
>> Mary`s[28] and Ann`s[26] ages compined total 44 years. Mary[28] is twice
>> as old as Ann[14] was when Mary[27] was half as old as Ann[54] will be
>> when Ann[54] is three times as old as Mary[18] was when Mary[18] was three
>> times as old as Ann[6]. How old Ann is?
>
>This is a much more satisfying answer and I think more along the
>lines of what I would have expected from a Sam Lloyd puzzle.
>--
>Stephen H. Landrum VOICE: (415) 813-8909
> Domain: slan...@ntg.com UUCP: ...apple!ntg!slandrum
> USNAIL: New Technologies Group Inc. 2468 Embarcardero Way, Palo Alto CA 94303
So sorry. Ann was 15 when Mary was 27 with your numbers.
Daniel Lance Herrick
>
>Solution: It's in the middle of a pool game (either snookers or eight ball, but
>since its in London, i would guess the former). Two billard balls have just
>hit each other, and the man (with his cue stick), knocks the cue ball ("pale")
>away...
I thought the appointments and time pointed at three cushion.
dan
Stephen H. Landrum
|>For my solution, I get Mary at 28 years, and Ann at 16. Mary is
|>more than 12 but less than 13 years older than Ann.
|>
|>> Mary`s[28] and Ann`s[26] ages compined total 44 years. Mary[28] is twice
|>> as old as Ann[14] was when Mary[27] was half as old as Ann[54] will be
|>> when Ann[54] is three times as old as Mary[18] was when Mary[18] was three
|>> times as old as Ann[6]. How old Ann is?
|>
|>This is a much more satisfying answer and I think more along the
|>lines of what I would have expected from a Sam Lloyd puzzle.
|
Ann turned 15 while Mary was 27, but Ann was 14 when Mary turned 27.
The whole point of my solution is that Mary and Ann do not share the
same birthdate, but do speak of their ages in whole years (as do most
people I know). Many of my friends and I are the "same" age for about
3/4 of the year, but some of us are "a year older" for part of the
year.
larry_baum
G. L. Sicherman) writes:
>Now here's another one by Sam Loyd:
>
> They were in a magnificently decorated room in the West End
> of London. They approached each other from opposite directions.
> Presently they met, and careless of the fact that dozens of
> eyes were watching them, they kissed each other with a resound-
> ing smack!
> The meeting seemed to bring them perfect peace; but alas,
> alack! they had scarcely been side by side above twenty seconds
> when a man approached with the fire of battle in his eye. With
> cool insolence he raised the stick he carried, and then--oh,
> horror!--he struck a sharp, quick blow, and the pale one was
> sent spinning several feet away.
> The other neither screamed nor fainted. There was no heart-
> breaking, no resentment; not even a murmur was heard, because---?
>
>--
>Col. G. L. Sicherman
>g...@hrmso.att.COM
>
Sounds like a fancy clock mechanism for striking the hour.