## A rather difficult brain teaser

I received the following brain teaser by e-mail and posted it to my brain teasers page. I hadn’t yet figured it out and hoped someone else would help me out.

Well, it’s now exactly one month from when I received it, and yesterday I finally got the answer. I wouldn’t dream of ruining the fun of figuring it out for anyone else, so here’s the riddle sans answer.

A princess is as old as the prince will be when the princess is twice the age that the prince was when the princess’s age was half the sum of their present ages.

What are their ages?

In the end, I had to pull out ye ol’ linear algebra from college to figure out the answer. There are others ways, but I didn’t have much success with them.

I may post a few hints in the comments, depending on how difficult it is for others to figure out.

Spoiler Warning: Answers are in the comments, so if you want to figure it out yourself, don’t scroll down.

1. princess: 40
prince: 30

Comment by chad on November 22, 2006 @ 12:02 pm
2. It’s a bit messy… but using simple algebra… I’m sure there are much easier ways of accomplishing the solution.

Princess: a
Prince: b

0.5*(a+b) a – (0.5*(a+b)) = 0.5*a-0.5*b b – (0.5*a-0.5*b) = 1.5*b-0.5*a 3*b-a
(3*b-a) – a = 3*b-2*a b + (3*b-2*a) = 4*b-2*a a = 4*b-2*a or 3*a = 4*b b+10
0.5*b+5 b – (0.5*b+5) = 0.5*b-5 a – (0.5*b-5) = a-0.5*b+5 b = a-0.5*b+5 or a = 1.5*b-5

Now we have two equations:
3*a = 4*b and a = 1.5*b-5.

Substitute “a” in the first equation for the value of “a” from the second equation:
3*(1.5*b-5) = 4*b

Solving this equation, we get:
b=30

Since a = 1.5*b-5:
a=40

Comment by chad on November 22, 2006 @ 12:23 pm
3. I gave up! I don’t think that is simple algebra, Chad. Good job guys.

Comment by katie on November 22, 2006 @ 2:17 pm
4. i hope someone can help me. my niece has a riddle to be solved by friday! “leslie and lorna have the same parents. leslie and lorna look exactly alike. leslie and lorna are the same age, yet they are not twins. how is this possible?” please e-mail me if you know the answer, or if there is a free site that can give it to me thank you for your help in advance. angie

Comment by angie9616 on January 23, 2007 @ 1:07 pm
5. They are both 0

Comment by Charlie on February 14, 2007 @ 12:04 am
6. em… baldur’s gate 2
simple, 2{m-[f-(f+m)/2]}-f+m=f

Comment by Kyle on October 30, 2007 @ 5:13 am
7. Princess – 978564
Prince – 12

I used simple math to find the answer.

u = 7x/6 = 2rp + u =11rp/6 – 34ru + 783r

And that means they have snazzy pants, yo.

Comment by Yay Yourface on February 19, 2008 @ 8:12 am
8. Hello there

Although I’m mighty impressed with the algebra skills of all you guys, I must say that in the explanation of Chad I don’t quite understand how he’s getting all of the following equations:

0.5*(a+b) a – (0.5*(a+b)) = 0.5*a-0.5*b b – (0.5*a-0.5*b) = 1.5*b-0.5*a 3*b-a
(3*b-a) – a = 3*b-2*a b + (3*b-2*a) = 4*b-2*a a = 4*b-2*a or 3*a = 4*b b+10
0.5*b+5 b – (0.5*b+5) = 0.5*b-5 a – (0.5*b-5) = a-0.5*b+5 b = a-0.5*b+5 or a = 1.5*b-5

merely out of that riddle;

and neither did I completely get Dan’s explanation, since in the beginning he’s saying that:

“since the difference in their age will always be the same.”

But when looking at the possible solutions, the difference in their age is only 2 years when the Princess is 8 and the Prince is 6, while it’s already 20 years in his last possible answer. How can this be the start of the riddle solving then? Perhaps he meant percentagewise? Or was he referring to something else?

In any case, no disrespect for the two however, if it helped them solve the riddle, then congratulations!

The way I did it, is slightly different; more a trial and error. Of course I did have the five possibilities at hand, that were presented in the game itself, which narrowed my search down a lot, but with a decent .xls file, I’m sure I could try all possible numbers between 1 and 100 in just the same time.

Here it goes:

When you read the riddle, you can make up that it involves the ages of only two persons; therefore I suggest you first make two columns; one for the Princess and one for the Prince. Furthermore the riddle
speaks of their age in the present, makes a comparison in the future, and states something about their age in the past. Therefore I suggest you make three lines per column.

You can represent this as follows:

AGE Princess Prince
Past A B
Present C D
Future E F

Now let’s see what the riddle says about the relations of their age.

relation 1:
A princess IS (Present) as old as the prince WILL BE (Future)
therefore C = F

relation 2:
when the princess is twice the age (Future) that the prince WAS (Past)
perhaps not so clear if you read the fragment separately, but if you read it well this means E = B x 2

relation 3:
when the princessâ€™s age WAS (Past) half the sum of their PRESENT ages (Present).
therefore A = Â½ (C + D)

A bit difficult to represent all that, using just plain text, but here I try anyway:

AGE Princess Prince

Past A/2 = _ B x 2
\ /
_____+__/___
| / |
Present C __ / D
\ /
=
/ \
Future E _/ \____ F

I hope that’s clear enough. Now, the five possibilities in the game were:

1 The princess is 40, prince is 30;
2 The princess is 30, prince is 40;
3 The princess is 30, prince is 20;
4 The princess is 20, prince is 30;
5 Their both the same age (let’s use 50).

1 2 3 4 5
AGE P’ss P’ce | P’ss P’ce | P’ss P’ce | P’ss P’ce | P’ss P’ce
Past A B | A B | A B | A B | A B
Present 40 30 | 30 40 | 30 20 | 20 30 | 50 50
Future E F | E F | E F | E F | E F

And use relation 1:

1 2 3 4 5
AGE P’ss P’ce | P’ss P’ce | P’ss P’ce | P’ss P’ce | P’ss P’ce
Past A B | A B | A B | A B | A B
Present 40 30 | 30 40 | 30 20 | 20 30 | 50 50
Future E 40 | E 30! | E 30 | E 20! | E 50!

Whatever difference in age the Prince has (his Present vs Future), we can of course extract to the Princess her age, therefore:

1 2 3 4 5
AGE P’ss P’ce | P’ss P’ce | P’ss P’ce | P’ss P’ce | P’ss P’ce
Past A B | A B | A B | A B | A B
Present 40 30 | 30 40 | 30 20 | 20 30 | 50 50 Future 50 40 | 20! 30! | 40 30 | 10! 20! | 50! 50!

As you can see, in possibilities 2, 4 and 5, the FUTURE age of the Princess and that of the Prince turn out to be either SMALLER than or EQUAL to their present age, which is of course illogical and not
possible, therefore you can already rule out these three!

Now let’s see how relation 2 affects the schedule:

1 2 3 4 5
AGE P’ss P’ce | P’ss P’ce | P’ss P’ce | P’ss P’ce | P’ss P’ce
Past A 25 | x x | A 20! | x x | x x
Present 40 30 | x x | 30 20 | x x | x x
Future 50 40 | x x | 40 30 | x x | x x

Again, whatever difference in age the Prince has (his Present vs his Past), the same difference in years must be valid for the Princess as well, therefore:

1 2 3 4 5
AGE P’ss P’ce | P’ss P’ce | P’ss P’ce | P’ss P’ce | P’ss P’ce
Past 35 25 | x x | 30 20! | x x | x x
Present 40 30 | x x | 30 20 | x x | x x
Future 50 40 | x x | 40 30 | x x | x x

As you can see in possibility 3, the age in the past is the same as the age in the present (for the both of them); illogical and not possible one full-blooded Vulcan would say, therefore we’ll no further explore this this option either.

This leaves us only one, but just to make sure we’re 100% correct, we’ll test relation 3 on it and see if the Past age of the Princess is indeed half the sum of their current ages:

A = Â½ (C + D)
35 = Â½ (40 + 30)

No need to be a rocket scientist to figure out this is indeed correct! Believe me when I say that you can use this way of working, using the three relations, in whatever order you like, you’ll still end up with the correct result. It also works with many other numbers, as long as they have the same correlation (one to the other) as 40 has to 30. I mean like 0,20 is to 0,15, or as 8000 is to 6000. Of course, these are no valid ages unless you’re genetically related to Gilgamesh, but just trying to back up Dan’s earlier statement here.

Now, I know this whole explanation may sound and look a lot more elaborate than the two previous ones, but honestly, I think in the end it’s much simpler. No need for algebra, it’s easy to understand for all.

I hope this will be useful for future people arriving on this site.

Regards

Didier

Comment by Lord_DDA on February 27, 2008 @ 10:17 am
9. Aaaaargh! Too bad the makup of the formula and sheme alters when uploading, makes it a lot less clear. Try to insert some spaces yourself when copying this to a notepad file, and you should see something like this:

AGE…….Princess……Prince
Past…….A/2.=……_.B x 2.
…………….\…./……..
…………_____+__/___……
………..|……./….|…..
Present….C.__…/…..D…..
……………\./…………
…………….=………….
……………/.\…………
Future…..E._/…\____.F…..

…………1………..2………..3………..4………..5…..
AGE…..P’ss.P’ce.|.P’ss.P’ce.|.P’ss.P’ce.|.P’ss.P’ce.|.P’ss.P’ce.
Past…..35…25..|..x….x…|..30…20!.|..x….x…|..x….x…
Present..40…30..|..x….x…|..30…20..|..x….x…|..x….x…
Future…50…40..|..x….x…|..40…30..|..x….x…|..x….x…

Comment by Lord_DDA on February 27, 2008 @ 10:32 am
10. 30, 40

Comment by Mary on October 21, 2008 @ 7:42 pm
11. The result, as posted earlier, is any two ages where the prince’s age is 3/4 of the princess’s (my answer was 15 and 20, respectively).

My process produced a much shorter equation, though:

Where prince = x and princess = y:

We know the princess is older (or the same age) because of the phrasing of the question: “princess is as old as the prince will be”, so the difference of the two ages is (y-x).

The equation ends up as:

Princess = (The sum of current ages divided by two minus their difference) times two minus their difference

or:

y = ((x+y)/2 – (y-x)) * 2 – (y-x)

Which expands as:
y = (x+y) – 2(y-x) – (y-x)
= x+y – 3(y-x)
= x+y – 3y + 3x
= 4x – 2y
3y = 4x
x = 3/4 * y

So then any combination of their ages where the ages are in that ratio will work. For the first few integers:

x,y
3,4
6,8
9,12
12,16
15,20
18,24

etc.

Comment by Kit Grose on December 5, 2008 @ 2:53 pm
12. im only 11 and I get it it is simple algebra. I covered this when I was 9 hes 30 shes 40 duhh

Comment by lindsey on March 3, 2009 @ 5:16 pm
13. You are sure a math freak and you are a geanis as well.

Comment by cheyenne on August 20, 2009 @ 5:43 pm
14. Urm can 1 of u boys marry me?hahaha i need 2 become familier with maths i recieved my matric at the age of 12 an im currently studing 4 a b.a degree in psychology haha bt u guys r mathmatically inclined so i love that! haha

Comment by Crystal on October 13, 2009 @ 8:35 am
15. princess:20
prince:40

Comment by hannah on December 23, 2009 @ 6:19 am
16. Thanks for telling us the answer we were completely baffled! But I dont think it was simple algebra. Anyway thank you yiu brainboxes :P

Comment by Nia + Katie Williams on December 31, 2009 @ 2:44 pm
17. Ahhhhhhhhhhhhhhhhhhhhhhh im ripping my hair out!

Comment by Lilly Ray on January 22, 2010 @ 5:29 pm
18. is the prince and the princess 23? or im just really clueless…?>:D

Comment by clueless on February 4, 2010 @ 4:10 am
19. that was hard

Comment by shanice on March 16, 2010 @ 2:06 pm
20. i dont understand

Comment by shanice on March 16, 2010 @ 2:08 pm
21. though it was but wasnt 100% sure

Comment by anna on March 26, 2010 @ 11:17 pm
22. There’s actually quite an easy way to solve this using only basic algebra (personally, I hate the matrix method).

Without trying to absorb all of the information in this statement, think of it as a mathematical equation. Therefore everytime you see “when”, mentally replace it with an equals sign.

“A princess is as old as the prince will be [WHEN] the princess is twice the age that the prince was [WHEN] the princessâ€™s age was half the sum of their present ages.”

If you’ll notice, we now have three equations that are all equal to each other, but we’ll first need to set some parameters before solving them. You should also notice in the question that it refers to both the past and future (prince WILL BE // princess’ age WAS, etc), so this refers to an event in time relative their ages (meaning + a certain amount of years or – a certain amount of years).

Current age of prince = X
Current age of princess = Y
+ Years in future = P
– Years in past = Q

Equation 1: Y = X + P
Equation 2: Y + P = 2(X – Q)
Equation 3: Y – Q = 1/2(Y + X)

3 Equations, but 4 unknowns implies that there are multiple solutions. Let’s set everything equal to 0 first (and multiply Equation 3 by two to make it easier). I’ve skipped a couple of steps by doing this, but you should get the same results if you follow what I’m doing.

Y – X – P = 0
Y + P + 2Q – 2X = 0
Y – 2Q – X = 0

We’re looking for the age of the prince and age of the princess (X and Y), so we want P and Q to equal zero. Instead of using a matrix, I took advantage of the convenient situation (look at the values of P and Q in these 3 equations – what happens when you add them?) and added all the equations together.

Therefore: 3Y – 4X = 0

Either: (1) y = 4/3X OR (2) X = 3/4Y

Which in English means: The prince is 3/4 the age of the princess :) (You can test this with any numbers for the original 3 equations, and you’ll always get the same P/Q ratio for the 3 equations – try it with X = 3, Y = 4)

Comment by Parris on April 5, 2010 @ 1:28 pm
23. Easier yet (well to me) using simple algebra.
H = Princeâ€™s present age
S = Princessâ€™s Present age
âˆ† = Difference in their ages and is always constant âˆ† = S – H or H = S – âˆ†
Breaking the problem down into understandable segments
princessâ€™s age was half the sum of their present ages. The princess at that time is Â½(S + H)
prince was (when the princessâ€™s age was half the sum of their present ages.)
The Prince was âˆ† less than the Princess so at that time the Princeâ€™s age is Â½(S + H) – âˆ†
princess is twice (the age that the prince was when the princessâ€™s age was half the sum of their present ages.)
Princessâ€™s age at that time is 2(Â½(S + H) – âˆ†)
prince will be (when the princess is twice the age that the prince was when the princessâ€™s age was half the sum of their present ages.)
The Prince again is âˆ† less than the Princess so at that time the Prince is 2(Â½(S + H) – âˆ†) – âˆ†
Finally
A princess is as old as (the prince will be when the princess is twice the age that the prince was when the princessâ€™s age was half the sum of their present ages.)
S= 2(Â½(S + H) – âˆ†) – âˆ†
H = 3âˆ†
Using H = S – âˆ†
S – âˆ† = 3âˆ†
S = 4âˆ†
Thus any combination of numbers where the Princeâ€™s age is 3 times and the Princessâ€™s age is 4 times the difference in their ages will work.

Comment by Josh on April 24, 2010 @ 9:38 am

and please dont leave the comment here email me on praptisarkar1@gmail.com please the riddle is

a man rides to a village on friday

stays there for three day and cmes back on friday

how is this possible??

please email me on the above id

Comment by prapti on April 30, 2010 @ 5:40 am
25. you are correct i mail you dan

tel me one thing do you receive mails not connected to the site tell me please

ok now tell me using this place cant we chat with each other

poor hav this, rich need this,if you drink or eat this you will die,if you wear this you will be embaresed tel now

a name of a vehicle when spelled backwards it remains the same

now what i say those anwers you line up together and form a sentence

say what is in up of your lips
say what is in your hand when it is empty

email me

Comment by prapti on May 3, 2010 @ 5:01 am
26. Sorry dan but your email has not reached me

Comment by prapti on May 9, 2010 @ 6:39 am
27. Bye the way when do you reply to these mails i send you

Comment by prapti on May 9, 2010 @ 6:42 am
28. wHAT DO YOU MEAN I MNT THAT YOU DID NOT REPLY ME IN MY INBOX

Comment by prapti on May 11, 2010 @ 5:09 am
29. mail me

Comment by prapti on May 11, 2010 @ 5:10 am
30. fine

whatever but from now you are not mailing me you are just going to write here ok? and you dint reply to my comment no.33

the riddle was there you have to reply that

my name is prapti not @prapti

so can you please call me prapti

Comment by prapti on May 12, 2010 @ 4:18 am
31. Can you explain #33

Comment by Star on May 13, 2010 @ 8:34 pm
32. @prapti youre funny
@dan good sport man!

Comment by mortified on August 7, 2010 @ 9:30 pm
33. so the prince is 3 and the princess is 4?

Comment by leo on August 29, 2010 @ 6:08 pm
34. Wow that was really hard it took me 3 days to waork it out :)

Comment by Tazza on September 18, 2010 @ 1:16 am
35. ;) wink :I hmmm that was hard
O(o_o)0 monkey hehehe
hard brain teaster though

Comment by Stuffed on September 18, 2010 @ 1:25 am
36. yup that’s a bit difficult.but still i am trying 2 get the ans

Comment by i think u can get it by my id on October 4, 2010 @ 9:49 am
37. wow!!

in all of your answers & reasoning, i had much thinking about the way people understands..the way they responds unto something..the way they reason out..the way their brains had developed into something beyond my learning as a student.

thank you!!

more power!!

once again,

WOW!!

Comment by vendetta on October 5, 2010 @ 10:58 pm
38. will some one please tell me what the simple answer is
im in 8th grade algebra and im 13
im of course are one of those clueless people that you all know and hate
i’ll get 10 extra points on a chapter test, and i really want an A+ to save my life
for the love of me please tell me….

Comment by maddy on October 13, 2010 @ 5:26 pm
39. Its simple algebra the princess is 40 and the prince is 30 (Use Algebra!)

Comment by Jack on November 6, 2010 @ 1:12 pm
40. As the puzzle is phrased at present, there is an infinite number of correct answers. I suggest saying, “A princess is as old as the prince will be when the princess is twice the age that the prince was when the princessâ€™s age was half the sum of their present ages. How old will the prince be when the princess is 40 years old?” This gives a single correct answer (30 years) without altering the original format of the puzzle.

Comment by Frank on November 7, 2010 @ 6:06 am
41. @ Frank: How the hell did u come up with the idea that stating the princess’s current age will single out an answer? :P
The prince’s age will remain a function of their age ratios regardless of the princess’s age.
Prince’s Age = 40 – 2^n, for n={0,1,2,3,…}
note that n depends on the age ratio.
Nice try though ;)

Comment by Mike on November 20, 2010 @ 6:04 am
42. @ Dan: How old will the prince be when the princess is 40 years old?
Equivalent to: How old will the prince be BY THE TIME the princess is 40 years old?

Answer: Depends on what 2 numbers u pick to satisfy their CURRENT 4 to 3 age ratio. I quote u here Dan:

“…This means that you can pick any difference you want (an even one presumably because you want integer ages).

Princess age: 4d
Prince age: 3d

Ages that work
Princess Prince
4 3
8 6
16 12
24 18
32 24
40 30
48 36
56 42
64 48
72 54
80 60”

*** Say you pick 16 for the princess and 12 for the prince, now you tell me Dan, HOW old will the prince will be by the time our little 16 year old princess reaches the age of 40?***

Note how the “extra” information stated in franks question doesn’t regard the princess’s CURRENT age, giving us the mere information that the princess will reach the age of 40 at some point in her life. It doesn’t give us a single answer simply because it doesn’t give any new information.

Now add something like: “The princess is CURRENTLY 40, how old does that make the prince?” and u are more than welcome to substitute 40 for the princess’s age, which yields 30 for the prince’s age. ;)
Or u can say: ” The princess will be 40 years old in 10 years, how old will the prince be by then?”
Or” The princess will be 40 years old in 10 years, what’s the current age of the prince?”
Although all of these questions I find naive, since u might as well save yourself all the trouble and give out the age of the princess right from the start.
In case u have read my reply up to here and have not yet come to my conclusion, I suggest re-reading Frank’s exact question once more and reconsidering your conception of the problem.

Comment by Mike on November 23, 2010 @ 9:57 am
43. U didn’t read the rest did u.

Comment by Mike on November 23, 2010 @ 1:51 pm
44. @Dan, I will explain again, this time in different words.
PART ONE:
Let’s consider the initial question:
A princess is as old as the prince will be when the princess is twice the age that the prince was when the princessâ€™s age was half the sum of their present ages.

What are their ages?
To take advantage of the information in this question, let’s denote M and F to the current male and female ages respectively. Hence we have 2 unknowns and one algebric equation which ultimately yields the following equation: F/M = 4/3 (remember f and m are ages, CURRENT ages to be precise)
Up to here we both agree that it is impossible to determine the numerical values of F and M and that we still need one more restraint to completely solve the equation and find the absolute values of F and M.(right? lol)
One simple way to provide the extra information needed to solve this problem would be to give the value of either F or M, and by substituting in the equation F/M = 4/3, the other unknown would be found.
At first this seems like precisely what Frank has done, finishing the problem with this question: “How old will the prince be when the princess is 40 years old?”
And u figured u can substitute 40 for F and yield 30 for M, right?
Do u agree with me up to here? Everything I said up to here we will refer to as part one from now on. Do u agree with part one of my argument?

Comment by Mike on November 23, 2010 @ 2:49 pm
45. PART TWO:
Let’s first review part one:
First of all, we name the current ages of the princess and prince F and M respectively, therefor the princess is currently F years old and the prince is M years old.
Also, we agree that the current age of the princess is 4/3 times bigger than that of the princes, in other words: F/M = 4/3
Last but not least, we agree that Frank ended his question with the following statement: “How old will the prince be when the princess is 40 years old?”, meaning that the princess is currently not aged 40, in other words F does NOT equal to 40. Note that F is the current age of the princess, and Frank’s statement says absolutely nothing about the princess’s current age.

Comment by Mike on November 23, 2010 @ 3:10 pm
46. “Jack and Jill are siblings, Jill is currently 16 years old and jack is 10, how old will jack be when Jill is 20?”
P.S. I am this close to killing myself lol

Comment by Mike on November 23, 2010 @ 3:23 pm
47. @ Dan: â€œJack and Jill are siblings, Jill is currently 16 years old and jack is 10, how old will jack be when Jill is 20?â€

My understanding of this statement is that Jill is currently 16, but she will be 20 someday. So Jill’s current age, F, is equal to 16 regardless of the fact that she’s going to be 20 someday. F = 16 not 20.

Comment by Mike on November 24, 2010 @ 1:46 am
48. Wow! I just visited this site ever since… I don’t like riddles exactly but then I love to see how people think deeply. I love to play chess and I am a national player before… but I am amazed with you guys! You are really good in Algebra huh!You really proved that men are analytical and logical. I enjoyed reading your posts! I hope you will make more.

Comment by mat on November 24, 2010 @ 6:17 am
49. @ Dan: Dan, yes of course, Frank’s version of the question was a trick question that could easily be misleading. When solving a problem u have to recognize what parts of information are useless and irrelevant and sort them out from the rest of the problem.
If u had carefully read my earlier post (#57), I already gave some versions of Frank’s question that would yield single answers, and there I stated that it would be a silly riddle if u gave out the princess’s current age, e.i. stating that “the princess is currently 40” after going through all the trouble to “explain” her age in a subtle and complicated way.

The problem goes on and on about the princess’s age, to the extent where it even relates her age to “the sum of their present ages”:

“A princess is as old as bla bla bla when the princessâ€™s age was half the sum of their PRESENT AGES.”

And then suddenly u wanna enclose the statement with: “gees, did I mention our princess if 40?” ??? lol
The problem suddenly becomes so easy, because the first thing that probably pops to the average mind is to substitute the precious number for the princess’s age and work the problem backwards to solve for the prince’s age (the only unknown).
Anyway that was one long argument we had lol. I hope I proved my point up to here, nevertheless u can disagree if u wish.

Comment by Mike on November 25, 2010 @ 4:33 am
50. Yeah, u might as well start off the question with:
“The princess’s age(which btw is 40), is that of the prince when the princess is twice the age that the prince was when bla bla bla”
LMAO

Comment by Mike on November 26, 2010 @ 11:28 am
51. wow! it is great i want to give the problem on my mathematician friend.

Comment by mark on November 27, 2010 @ 11:49 pm
52. Your algebratic equations are commendable. Consider this now.
If it takes a chicken and a half. To lay an egg and a half. In a day and a half. How long would it take a monkey with a wooden leg to kick the seeds out of a watermelon. Answer at stricon@sympatico.ca on request. Only intelligent answers or questions accepted.

Comment by Grandmaster Jay on December 23, 2010 @ 1:19 am
53. i think it would be 30 & 40 i am i right !!!!!!!!!!!uffffff…..so tough ya

Comment by aysha on February 28, 2011 @ 8:05 am
54. weell that is so simple,try to ask the one who made that question

Comment by anna on March 14, 2011 @ 6:20 am
55. Is the question complete?

Comment by koushik on March 15, 2011 @ 2:18 pm
56. I’m 11 so i just went simple and said they where as old as they are

I DONT KNOW ALGEBRA!!! (OR AT LEAST NOT MUCH)

Comment by damen on May 26, 2011 @ 5:53 pm
57. Let X = current age of princess, Y = current age of prince.

There are three moments of time that are significant for the problem:
T1 = when princess’ age is half their current ages;
T2 = current time;
T3 = when the princess is twice the age that the prince was at T1.

Since princess is X years old now (at T2) and (X + Y)/2 at T1, the moment T1 was X – (X + Y)/2 = (X-Y)/2 years ago.

The prince, who is presently age Y, was age Y – (X-Y)/2 = (3Y – X)/2 years old at T1 (subtract from his current age the number of years that passed between T1 and T2).

At T3, princess’ age will be twice the age of prince at T1, i.e., 3Y – X. That means that 3Y – X – X = 3Y – 2X years passed from T2 to T3.

Add that to prince’s current age to get his age at T3, i.e., at T3 prince is 3Y – 2X = 4Y – 2X years old.

The main and only condition of the riddle is that princess’ current age is the same as the age of prince at T3, i.e.,
X = 4Y – 2X or 3X = 4Y or Y = 3X/4

Also, T1 was (X-Y)/2 = X/8 years ago; and T3 will be 3Y – 2X = X/4 years from now.

To summarize,
(T1) X/8 years ago princess was 7X/8 and prince was 5X/8 years old;
(T2) Currently, princess is X and prince is 3X/4 years old; and
(T3) X/4 years from now they are 5X/4 and X years old.

I believe that all the numbers in this write up should be integers, otherwise you will end with statements like
“Prince and princess are 3 and 4 years old; half a year ago princess’ age was half of the sum of their current ages (3.5) …” – to me that simply does not sound like a reasonable statement about numbers that express human ages.

The highest denominator in the description above is 8, so all numbers will come out as integers if X is divisible by 8. So, my possible answers for the current ages of princess and prince are:
8, 6
16, 12
24, 18
32, 24
40, 30 etc. until 128, 96 (no need to go further as there are no known humans aged more than 130; in fact, one may claim that the problem implies that the age of princess at T3 should also be plausible, in which case her current age should not exceed 104).

In comparison to some other solutions offered I reject solutions such as 4,3 or 12,9 etc. because they imply that some important ages and/or time intervals in the problem are expressed as “half years” rather than integers.

Comment by Alex on June 10, 2011 @ 8:27 am
58. i dont no wat age they are but i no that wen they ar ehalf their age they were 1 and 2!!!

Comment by smiley rylie on June 12, 2011 @ 1:44 am
59. Their ages are their ages,,.. Thank you Thank you!

Comment by Princess Maricar on July 2, 2011 @ 4:48 am
60. The princess is X the prince is Y
y=presentX*2 the age as the pastY=pastX when X was half of the parents age combined(70=parents combined age)(35=half of parents combined age=pastX=pastY)(70=pastY double age=presentX=presentY)
Wouldn’t the prince and the princess be the same age?
Prince=70
Princess=70

Comment by Cody on July 15, 2011 @ 11:48 am
61. Sorry I read wrong not parents it was present

Comment by Cody on July 15, 2011 @ 11:54 am
62. Sorry they 40 and 30 not 70 i did a mess up

Comment by Cody on July 15, 2011 @ 11:56 am
63. The answers are 30,40 or 20,60 or 40,40

Comment by Cody on July 15, 2011 @ 12:03 pm
64. And I am only 11 lololololololololololol

Comment by Cody on July 15, 2011 @ 12:05 pm
65. Oh ok

Comment by Cody on July 16, 2011 @ 1:00 pm
66. shes 30 hes 40

Comment by jake on July 19, 2011 @ 3:58 pm
67. Princess:prince=4:3

Comment by Arnab on July 28, 2011 @ 5:10 am
68. that was very hard to do :( but when i seen the answer and seen the working out i knew than it was not as hard :P

Comment by Louise on August 14, 2011 @ 11:50 am
69. Gud day..
I have 1 question that bothers me.does the riddle saying “when the princess is twice the age that the prince was” means comparing the future age of the princess (is twice) to past age of the prince(the prince was)?
I think we should compare the present age of the princess to that past age of the prince.
Thank you.
Fr.ph

Comment by nayr1001 on August 16, 2011 @ 11:04 pm
70. This question is not to be answered. If we are wise we will leave it !!!

Comment by Abhinay on August 29, 2011 @ 2:53 am
71. da ans r horrible nd i didnt understood any
:)

Comment by nv on September 25, 2011 @ 2:52 am
72. The answer is 3y=2x under condition that x is an even number, multiple of 3 and is greater than 6… try x=12 and y=8
where y is the age of the boy and x is the age of the girl.

Comment by Arvesh on October 19, 2011 @ 2:47 am
73. THEY ARE BOTH over 9000 so we must kill the batman. also the princess is a man.

Comment by chickenbutt on October 26, 2011 @ 10:57 am
74. the princes is: 4 while the prince is: 2
when the princes was half the sum of their ages she was 3: while the prince was two years younger so he was: 1 .
if we multiply the prices age at that time by two it will give us the answer 2 which is the prince’s age now!
please comment if It sounds that I understood the question wrongly!
thanks

Comment by 00100 on November 14, 2011 @ 4:11 pm
75. I think i miss understood the question at the beginning, anyway the correct answer is the princess:40 and the prince is: 30 .
when the princes was half the sum of their ages she was 35 while the prince was 25 at that time, and twice that is 50, so when the princes is 50 the prince will be 40 (10 years less) which is her current age.

Comment by 00100 on November 17, 2011 @ 10:10 am
76. do it like this

2(((a+b)/2)-(a-b))-(a-b)=a
or twice as much as (half their ages)-(the difference in their ages) – (the difference in their ages)=(her age)

http://www.wolframalpha.com/input/?i=%282%28%28%28a%2Bb%29%2F2%29-%28a-b%29%29-%28a-b%29%29%3Da

Comment by tobias on December 1, 2011 @ 2:42 pm
77. Wow. Im 11 years old and i figure stuff like this out all the time. My personal guess is this:::

PRINCESS- 40
PRINCE- 30

Ok, if thats wrong, then please tell me someone I mean come on im only 11

Comment by makayla on December 7, 2011 @ 12:58 pm
78. Let me give u all a clue i’ve gÃ³tten the answer but i want u to try it.To answer dis ? It should start 4rm present, past, and future u will get it.

Comment by ahmad ddon on December 25, 2011 @ 11:04 am
79. I GIVE UPPPP!after all it’s not lyk the answer its the code-word to heaven,,or is it??????

Comment by jonni on January 10, 2012 @ 1:16 am
80. they are both the same age. instead oif cheaking your work by plugging varibles back into the equation, try checking it with the actual word problem.

A princess is as old as the prince will be when the princess is twice the age that the prince was when the princessâ€™s age was half the sum of their present ages.
so if the price and princess are both zero the riddle still works this should work if you solved it algebraically.

those who said it doesnt matter what ages they were just as long as it followed a 4:3 ratio are wrong because the riddle deals with past present or future and as soon as they get a year older the ratio would no longer be present. ex. the princess is 40 and the prince is 30 woulnd work because one year later they would be 41 and 31. that is why they must be the same age because then even after they get a year older their ages will still follow the same ratio.

Comment by thanksgiving on February 4, 2012 @ 11:44 am
81. Everyone doing this in such a heavy-handed way. You can solve with just 2 variables, the Princes age, and the difference between their ages (diff). Read it through and form the simultaneous equation, putting the equals before “when the princess was half the sum…”

(Prince + diff + diff)/2 + diff = (Prince + Prince + diff)/2.

Double both sides

Prince + (4 x diff) = (2 x Prince) + diff

Prince = 3 differences.

So we can’t know their ages, only that the Princess is older than the Prince by 1/3 his age. I like the riddle, the question should be “what can we know about their ages?”

Comment by Tom on February 19, 2012 @ 3:59 pm
82. Ans -4 is very complicated you can solve it very easily by using algebraic equations.

x-y=d
2z-x=d
(x+y)2 – z =d

y=3d
x=4d

Comment by babu on March 13, 2012 @ 2:52 am
83. the princess’s age in the past – the difference in their ages = the prince’s age in the past
the prince age in the past *2 = the princess’s age in the future
the princess’s age in the future – the difference in their ages = the prince’s age in the future , which is the princess’s age in the present
so if the princess’s age is a and the prince’s age is b
a = ((((a+b)/2)-(a-b))*2 )-(a-b)

Comment by sameh on March 22, 2012 @ 11:06 am
84. I’m extremely miffed, I was trying to do this riddle in my head and came up with 4 & 3, 8 & 6, 16 & 12 but couldn’t quite tie it all together. I didn’t think we were ‘allowed’ to use a pen and paper!

Comment by Scott on April 14, 2012 @ 11:53 am
85. Myself of course! Haha.

Comment by Scott on April 15, 2012 @ 6:48 am
86. Hey lets say that the princess is 1 year old, and the prince will be as old as the princess when she is twice as old as 1 year but half of her current age. So the princess would be 2 years old and the prince is 1 year old. Maybe they are brother and sister.

Comment by Charlie on May 11, 2012 @ 12:09 am
87. Nevermind it wouldn’t work with the princess being a year older they would have to be the same age for it to work, that way when the princess gets a year older and with the riddle in mind it would go.
princess: 2×3=6 6\2=3
The prince and princess would be the same age with that kind of math as that’s what the riddle says times the age by two as that is twice the age, and then divide by two and that is half the age.

Why make the riddle so difficult by doing harder math? All you had to do was multiply by 2 then divide by 2 by using any number it would all be the same answer. The princess and prince are the same age.

Comment by Charlie on May 11, 2012 @ 12:29 am
88. as said princess age is twice as prince age. so let x be prince age. so princess will be 2x.
prince = x
princess = 2x. princess age was half of their present ages. so

princess = (2x+x)/2 = 3x/2 = 1.5x

so if x(prince) age is 2, princess age is 3.
if prince age is 10, princess age will be 15. so on…

Comment by surendra on July 13, 2012 @ 8:17 am
89. I think most people simply complicate things. It all boils down to a system of three linear equations:
x = y + a
x + a = 2 (y â€“ 1/2a)
x â€“ 1/2a = (x + y)/2

where x is the princessâ€™ age, y is the princeâ€™s age, and a is the age difference between them. The system is then easily solved by repeatedly eliminating variables, or
using Gaussian elimination, no need for matrixes really.
The result is:
X = 4a
Y = 3a

The solution set is determined by assigning values to the variables such that each of the equations is satisfied.

Comment by stella on August 7, 2012 @ 3:19 pm
90. So if there’s more than one possible correct answer, then this is silly. It may encourage us to think but this is more of Mathematical probability or problem solving rather than a mere riddle or brain teaser. Because it could be 8 and 6 or 40 and 30. The thing is, if I will answer 8 and 6, I can no longer consider the 40 and 30. Because 8 and 6 has only 2 years interval, and when my princess reaches 40 and the prince is 30 that’s a 10-year interval and both are considered as current. It’s no longer the same princess and prince I have for my subject. I hope you’re getting my point. Lets say my answer is 8 and 6 and someone will claim that 40 and 30 is also correct, then by the time my princess turns 40, my prince should be 38…

See, the question really needs to provide a year at least, because not providing a year for any event, will come up with multiple correct answers. But if we are talking about only one princess and a prince, there would be no correct answer.

Ages can not be determined as to “either this or that”. As we could only have one age. I can not be 30 or 40 hahahaaaa

Comment by dusk on August 13, 2012 @ 5:24 am
91. The riddle should have had asked, what is the youngest valid age the princess and prince can have? For a valid age is one (1 year old) so we can come up with only one answer.

Comment by dusk on August 13, 2012 @ 5:34 am
92. The answer is always like anything possible to 4/4 and 3/4 for the princess and prince respectively… regardless… meaning the answer can be this can be that as long as the princess age can be 1/4 greater than the prince. But the question is what is the age of the princess and the prince as if there is only one answer!!!

You see?
Princess Prince
4 3
8 6
16 12
24 18
32 24
40 30
48 36
56 42
64 48

That’s some of the possible answers but that will also give possible sets of princess and prince. not just one princess and one prince as their age interval changes. But in reality, if these are Mary and John, when Mary was 4 and John was 3, after years, Mary can’t be 64 and John is 48! How come?

So my point is the riddle is not appropriate. Thanks

Comment by dusk on August 13, 2012 @ 5:45 am
93. Let A = the present age of the princess.
Let B = the present age of the prince.

(1) a certain number of year in the future,
and
(2) a certain number of years in the past.

Let x = the number of years in the future that is spoken of.
Let y = the number of years in the past that is spoken of.

A princess is (now) as old as the prince will be
(in x years).

Therefore

A = B + x

In x years, the princess will be twice the age that
the prince was y years ago.

Therefore,

A+x = 2(B-y)

And y years ago, the princess was half the sum of their
present ages:

We have 4 unknowns but only 3 equations. So we
suspect that there may be more than one solution.

Simplifying those

A – B – x = 0
A – 2B + x + 2y = 0
A – B – 2y = 0

That give the matrix:

[1 -1 -1 0 | 0]
[1 -2 1 2 | 0]
[1 -1 0 -2 | 0]

which simplifies to rref form:

[1 0 0 -8 | 0]
[0 1 0 -6 | 0]
[0 0 1 -2 | 0]

A – 8y = 0
B – 6y = 0
x – 2y = 0

A = 8y
B = 6y
x = 2y

(A, B, x) = (8y, 6y, 2y)

The first solution that makes sense is when y=1
Then A=8, B=6, x=2, and y=1

The princess’ present age is 8
The prince’s present age is 6
The number of years in the future talked about is 2
The number of years in the past talked about is 1.

Checking:
The sum of their present ages is 14.
Then half the sum of their present ages is 7.
And indeed, the princess was 7 1 year ago.
In x=2 years the Princess will be 10, and
that is twice the age the Prince was y=1
year ago, for he was 5 then.

The next answer is when y=2

(A, B, x) = (8y, 6y, 2y)

The next solution that is when y=2
Then A=16, B=12, x=4, and y=2

The princess’ present age is 16
The prince’s present age is 12
The number of years in the future talked about is 4
The number of years in the past talked about is 2.

Checking:
The sum of their present ages is 28.
Then half the sum of their present ages is 14.
And indeed, the princess was 14 2 years ago.
In x=4 years the Princess will be 20, and
that is twice the age the Prince was y=2
years ago, for he was 10 then.

Comment by xguru on October 19, 2012 @ 4:57 pm
94. I am 9 years old in dubai in uae.i am an indian

Comment by Glossypink on November 26, 2012 @ 12:09 am
95. dis brain teaser is difficult…

Comment by riya on November 27, 2012 @ 3:51 am
96. I finally understood aft reading sameh’s method which is comment 107..Im 11!!Hahaha

Comment by Jasmine on December 6, 2012 @ 6:01 pm
97. I find it amusing that you all explain it instead of just tellng us. I’ve been reading all of the algebra and I still don’t know the answer! BTW I’m only 10 but still.

Comment by Jayjay on December 11, 2012 @ 3:41 pm
98. Can you help find the answer to 1+1 because I’m having way to much trouble on that. What does it answer!!!!

Comment by Abby on December 21, 2012 @ 1:41 pm
99. Did you know that 1+1=11, 2+2=fish, 3+3=8.

Comment by Abby on December 21, 2012 @ 1:44 pm
100. I know this is really old, but I’d go with Dan on this one. I don’t know how chad got a constant in his equation: 3*a = 4*b b+10. Where did that 10 come from. If you put the riddle in algebraic equations, you get 6 equations with 6 unknowns but no constants. If the princess is x years old and the prince is y years old, you can reduce the equations to y = 4/3x. Here’s the proof:

princess’s age = prince’s age
x = e (first relationship) eqn. 1
c = 2b (second relationship)eqn. 2
a = (x + y)/2 (final relationship) eqn. 3

age gap is constant:

x – y = a – b eqn. 4
x – y = c – e eqn. 5
a – b = c – e eqn. 6

from eqn. 4:

a = x – y + b

sub this in eqn. 1

x -y + b = (x + y)/2
2b = 3y -x

sub this in eqn. 2

c = 3y – x

sub this in eqn. 5

x – y = 3y – x – e

eqn. 3 is x = e

x – y = 3y – 2x

y = 4/3*x

To get integer values, any multiple of 3 will work.

Comment by Nemo on January 12, 2013 @ 9:45 pm
101. Actually, equation 6 is redundant. So this leaves 6 unknowns and 5 equations which means the system of equations is indeterminate. If there was a sixth equation with no constants, a single solution will exist, albeit an iterative method would have to be used.

Comment by Nemo on January 13, 2013 @ 1:58 am
102. I cant give up..but still accept ur workings (xguru)..with 57%

Comment by seun on February 13, 2013 @ 5:58 pm
103. I have an easier idea just :

2. Look at the examples of the other people.
3. Then get the anwser off of them.

See really easy, but cheating

Comment by amyrose on February 28, 2013 @ 8:17 pm
104. Guys..This one is rateher tricky,though answer lies in first part only. That is “Princess is as old as Princess”. That means they are twins, and the simplest answer to the question that what r their ages? is “Equal”.

Here it is.. how?
=================
Let us break it in parts…
A princess is as old as the prince will be…..(1)
prince will be when the princess is…..(2)
princess is twice the age that the prince was….(3)
prince was when the princessâ€™s age was half the sum of their present ages…..(4)

all (1)to (4) above indecates equalities with different timestamp.
so we can skip going to the loop of (2) and (3)
and get to conculsion by using (1) and (4)

A princess is as old as the prince will be, “%- prince was -%” when the princessâ€™s age was half the sum of their present ages.

so… they are twins… and this question will stand ture for any age you take.

what are their ages?
ANS :- Their ages are Equal.

Comment by Ankur Khakhar on March 20, 2013 @ 11:07 am
105. Sorry I missed out one step in the explaination. Here is modified verion.

Guys..This one is rateher tricky,though answer lies in first part only. That is â€œPrincess is as old as Princeâ€. That means they are twins, and the simplest answer to the question that what r their ages? is â€œEqualâ€.

Here it is.. how?
=================
Let us break it in partsâ€¦
A princess is as old as the prince will be whenâ€¦..(1)
prince will be when the princess isâ€¦..(2)
princess is twice the age that the prince wasâ€¦.(3)
prince was when the princessâ€™s age was…(4)
Princess’s age was half the sum of their present agesâ€¦..(5)

all (1)to (4) above indecates equalities with different timestamp.
so we can skip going to the loop of (2),(3) and (4)
and get to conculsion by using (1) and (5)

A princess is as old as the prince will be when, the princessâ€™s age was half the sum of their present ages.

soâ€¦ they are twinsâ€¦ and this question will stand ture for any age you take.

what are their ages?
ANS :- Their ages are Equal.

Comment by Ankur Khakhar on March 20, 2013 @ 11:14 am
106. How is that, completely wrong??

Comment by Ankur Khakhar on March 20, 2013 @ 11:19 am
107. WTF I DONT GET IT!!! XD

Comment by April on April 17, 2013 @ 5:22 am
108. I do not think it would be possible to state their exact ages in the present, but what is plausible is to state the current relation between their ages.
Somewhat like this:

Let the princess’ current age be X and the prince’s Y.

I chose to start backwards: “when the princess’s age was half the sum of their present ages.” This is a situation in the past, so
Princess(past) = 1/2(X+Y)

As the difference in their ages is constant, a current difference of X-Y would also apply in the past. Note here that X-Y is being considered because the princess is older. The riddle starts with stating that the princess IS how old the prince WILL BE.

Therefore,
Prince(past) +(X-Y) = Princess(past)

“when the princess is twice the age that the prince was”. This situation hints to the future (“when the princesss is”). So,
Princess(future) = 2 x Prince(past)

Therefore, as we know the difference of their ages,
Prince(future) + (X-Y) = Princess(future)

“A princess is as old as the prince will be”. This indicates that the princess’ present age must be equated to the prince’s future age.
Therefore,
Princess(present) = X = Prince(future)
=> X = Princess(future) – (X-Y)
=> X = {2 x Prince(past)} – (X-Y)
=> X = {2 x (Princess(past) – (X-Y))} – (X-Y)
=> X = [2 x {(1/2(X+Y)) – (X-Y)}] – (X-Y)
=> X = [2 x {3Y/2 – X/2}] – (X-Y)
=> X = [3Y – X] – (X-Y)
=> X = 3Y – X – X + Y
=> X = 4Y – 2X
=> 3X = 4Y
=> X = 4Y/3 ~ Y = 3X/4

Or, the Prince is currently 0.75 times the age of the Princess.

Comment by Prerna on July 18, 2013 @ 1:34 pm
109. PLS HELP ME FINE DIS:X IS A SEVEN LETERED WORD,X IS IMPOSIBLE FOR GOD.NEW BORN BABIES LIKED X BETTER DAN BREAST MILK,D POOR HAV X AND D RICH LOOK FOR X FROM D POOR,IF U EAT X U WL DID,X IS MORE IMPORTANT DAN LIFE.

Comment by VAL on July 26, 2013 @ 10:07 am
110. oh that was a hard one

Comment by anik on July 27, 2013 @ 1:57 am
111. I don’t get it but I just gonna try to understand it by ther equations cause I’m just a grade 6.

Comment by tracy lyka on August 6, 2013 @ 8:52 am
112. You also forgot that the sum of their ages must be even, because we have an integer problem here.

Comment by V on October 22, 2014 @ 1:10 pm