Fun little math problem
05-11-2016, 09:23 AM
Post: #21
 Tugdual Senior Member Posts: 744 Joined: Dec 2013
RE: Fun little math problem
(05-10-2016 09:43 PM)Massimo Gnerucci Wrote:
(05-10-2016 09:13 PM)Tugdual Wrote:  Why not? This is kind of a cal forum here...

Of course you may, but do you use one when you see 1+2+3=?

05-11-2016, 10:43 AM
Post: #22
 Massimo Gnerucci Senior Member Posts: 1,799 Joined: Dec 2013
RE: Fun little math problem
(05-11-2016 09:23 AM)Tugdual Wrote:

;)

Greetings,
Massimo

-+×÷ ↔ left is right and right is wrong
05-11-2016, 11:37 AM (This post was last modified: 05-11-2016 11:40 AM by Gerson W. Barbosa.)
Post: #23
 Gerson W. Barbosa Senior Member Posts: 1,211 Joined: Dec 2013
RE: Fun little math problem
(05-11-2016 07:57 AM)Paul Dale Wrote:
(05-11-2016 06:31 AM)Dieter Wrote:  How did you do it?

I figured the missing values were integers & it was then clear that they must be small. There were not a lot of options left: 2 & 4, 3 & 9, 4 & 16. I got it right first go after this.

It looks like you and Dieter have tackled the problem the same way it was done more than one hundred years ago by young Ramanujan:

"One afternoon back in 1902, during recess, an older student, said to be the smartest in his class, handed him a math problem. Ramanujan was so smart? Well, then, let him solve this: At first glance falling under the familiar heading of “two simultaneous equations in two unknowns,” the problem actually confronted Ramanujan with a difficult fourth-degree equation and meant recalling a theorem applicable to a particular class of them. To any ordinarily smart fourteen-year-old, it would be exceedingly difficult. “To my astonishment,” Rajagopalachari remembered later, “Ramanujan worked it out in half a minute and arrived at the answer by two steps.” In fact, he probably didn’t “work it out” at all, but simply looked at it, guessed the answer might be one where each was a square, tried a couple of possibilities in his head, and saw the solution, x = 9 and y = 4, jump out at him; in other words, it was a piece of fancy footwork, nothing mathematically profound." From "The Man Who Knew Infinity: A Life of the Genius Ramanujan (English Edition)" by Robert Kanigel.
05-11-2016, 01:07 PM
Post: #24
 Claudio L. Senior Member Posts: 1,645 Joined: Dec 2013
RE: Fun little math problem
(05-11-2016 07:57 AM)Paul Dale Wrote:
(05-11-2016 06:31 AM)Dieter Wrote:  How did you do it?

I figured the missing values were integers & it was then clear that they must be small. There were not a lot of options left: 2 & 4, 3 & 9, 4 & 16. I got it right first go after this.

Pauli

I had a similar path, slightly different though. From both equations I figured both x and y had to be perfect squares. Then looked at the first equation and y had to be less than 7, which left me with 4 as the only answer, leading to sqrt(x)=3. Visually tested the 2nd equation and turns out it worked.
05-11-2016, 02:23 PM
Post: #25
 Massimo Gnerucci Senior Member Posts: 1,799 Joined: Dec 2013
RE: Fun little math problem
(05-11-2016 11:37 AM)Gerson W. Barbosa Wrote:
(05-11-2016 07:57 AM)Paul Dale Wrote:  I figured the missing values were integers & it was then clear that they must be small. There were not a lot of options left: 2 & 4, 3 & 9, 4 & 16. I got it right first go after this.

It looks like you and Dieter have tackled the problem the same way it was done more than one hundred years ago by young Ramanujan:

"One afternoon back in 1902, during recess, an older student, said to be the smartest in his class, handed him a math problem. Ramanujan was so smart? Well, then, let him solve this: At first glance falling under the familiar heading of “two simultaneous equations in two unknowns,” the problem actually confronted Ramanujan with a difficult fourth-degree equation and meant recalling a theorem applicable to a particular class of them. To any ordinarily smart fourteen-year-old, it would be exceedingly difficult. “To my astonishment,” Rajagopalachari remembered later, “Ramanujan worked it out in half a minute and arrived at the answer by two steps.” In fact, he probably didn’t “work it out” at all, but simply looked at it, guessed the answer might be one where each was a square, tried a couple of possibilities in his head, and saw the solution, x = 9 and y = 4, jump out at him; in other words, it was a piece of fancy footwork, nothing mathematically profound." From "The Man Who Knew Infinity: A Life of the Genius Ramanujan (English Edition)" by Robert Kanigel.

Amazing: I didn't imagine it could be so difficult to see the solution. My previous comments stem from me doing almost the same, in a similar amount of time. And my brain is four times older than Ramanujan's.

I only wish I could have a glance to the other 99999 things he saw that I am still blind to...

Greetings,
Massimo

-+×÷ ↔ left is right and right is wrong
05-11-2016, 05:13 PM
Post: #26
 Gerson W. Barbosa Senior Member Posts: 1,211 Joined: Dec 2013
RE: Fun little math problem
(05-11-2016 02:23 PM)Massimo Gnerucci Wrote:  Amazing: I didn't imagine it could be so difficult to see the solution. My previous comments stem from me doing almost the same, in a similar amount of time. And my brain is four times older than Ramanujan's.

Quite easy indeed. So let's be mean, but not so mean: :-)

$$\left \{ _{\sqrt{y}+x=4}^{\sqrt{x}+y=2} \right.$$

05-11-2016, 05:24 PM (This post was last modified: 05-11-2016 05:28 PM by Massimo Gnerucci.)
Post: #27
 Massimo Gnerucci Senior Member Posts: 1,799 Joined: Dec 2013
RE: Fun little math problem
(05-11-2016 05:13 PM)Gerson W. Barbosa Wrote:
(05-11-2016 02:23 PM)Massimo Gnerucci Wrote:  Amazing: I didn't imagine it could be so difficult to see the solution. My previous comments stem from me doing almost the same, in a similar amount of time. And my brain is four times older than Ramanujan's.

Quite easy indeed. So let's be mean, but not so mean: :-)

$$\left \{ _{\sqrt{y}+x=4}^{\sqrt{x}+y=2} \right.$$

Zero-four
? :D

Greetings,
Massimo

-+×÷ ↔ left is right and right is wrong
05-11-2016, 05:26 PM
Post: #28
 Dieter Senior Member Posts: 2,398 Joined: Dec 2013
RE: Fun little math problem
(05-11-2016 05:13 PM)Gerson W. Barbosa Wrote:  Quite easy indeed. So let's be mean, but not so mean: :-)

OK, this time I could solve it in 30 seconds, too. ;-)

(05-11-2016 05:13 PM)Gerson W. Barbosa Wrote:  No numerical solutions, please.

What kind of solution do you expect here?

Dieter
05-11-2016, 05:28 PM
Post: #29
 Dieter Senior Member Posts: 2,398 Joined: Dec 2013
RE: Fun little math problem
(05-11-2016 05:24 PM)Massimo Gnerucci Wrote:  Zero-four?

Four zero.

Dieter
05-11-2016, 05:30 PM
Post: #30
 Massimo Gnerucci Senior Member Posts: 1,799 Joined: Dec 2013
RE: Fun little math problem
(05-11-2016 05:28 PM)Dieter Wrote:
(05-11-2016 05:24 PM)Massimo Gnerucci Wrote:  Zero-four? :D

Four zero.

Dieter

Too easy if you don't mix up... ;)

Greetings,
Massimo

-+×÷ ↔ left is right and right is wrong
05-11-2016, 06:14 PM
Post: #31
 Gerson W. Barbosa Senior Member Posts: 1,211 Joined: Dec 2013
RE: Fun little math problem
(05-11-2016 05:26 PM)Dieter Wrote:
(05-11-2016 05:13 PM)Gerson W. Barbosa Wrote:  Quite easy indeed. So let's be mean, but not so mean: :-)

OK, this time I could solve it in 30 seconds, too. ;-)

(05-11-2016 05:13 PM)Gerson W. Barbosa Wrote:  No numerical solutions, please.

What kind of solution do you expect here?

Dieter

Oops! I'd make for a good math teacher (for students). Kind of those who don't know how to set up a problem :-)

Ok, let's try again:

Find both solutions to

$$\left \{ _{\sqrt{y}+x=4}^{\sqrt{x}+y=2} \right.$$

Closed-form solutions only.
05-11-2016, 06:17 PM (This post was last modified: 05-11-2016 06:18 PM by Gerson W. Barbosa.)
Post: #32
 Gerson W. Barbosa Senior Member Posts: 1,211 Joined: Dec 2013
RE: Fun little math problem
(05-11-2016 05:30 PM)Massimo Gnerucci Wrote:
(05-11-2016 05:28 PM)Dieter Wrote:  Four zero.

Dieter

Too easy if you don't mix up...

One zero.

Massimo 1
Gerson 0

You win!
05-11-2016, 06:38 PM
Post: #33
 Tugdual Senior Member Posts: 744 Joined: Dec 2013
RE: Fun little math problem
(05-11-2016 05:13 PM)Gerson W. Barbosa Wrote:
(05-11-2016 02:23 PM)Massimo Gnerucci Wrote:  Amazing: I didn't imagine it could be so difficult to see the solution. My previous comments stem from me doing almost the same, in a similar amount of time. And my brain is four times older than Ramanujan's.

Quite easy indeed. So let's be mean, but not so mean: :-)

$$\left \{ _{\sqrt{y}+x=4}^{\sqrt{x}+y=2} \right.$$

I could reduce the problem to a 3rd degree equation and solved it with Wolfram.
Result is huge...
05-11-2016, 07:29 PM
Post: #34
 Gerson W. Barbosa Senior Member Posts: 1,211 Joined: Dec 2013
RE: Fun little math problem
(05-11-2016 06:38 PM)Tugdual Wrote:
(05-11-2016 05:13 PM)Gerson W. Barbosa Wrote:  Quite easy indeed. So let's be mean, but not so mean: :-)

$$\left \{ _{\sqrt{y}+x=4}^{\sqrt{x}+y=2} \right.$$