Fun little math problem

[David Hayden]

I was at a high school reunion this weekend and former teacher of mine gave me this little gem. It isn't hard to solve, but it's interesting that there is exactly one solution.

Find two positive numbers such that:

1. Adding them produces the same number as multiplying them
2. Subtracting them produces the same number as dividing them (although a different number from above)
3. Their difference is less than 1.
   

[Gerson W. Barbosa]

1+sqrt(2), 1+(sqrt(2))/2

[Tugdual]

I also have \(1-\frac { \sqrt { 2 } }{ 2 } \) and \(1-\sqrt { 2 } \)
Thanks 50G

Ooops sorry it was said positive numbers. Only one solution then.

[BarryMead]

(05-09-2016 08:29 PM)Gerson W. Barbosa Wrote:  1+sqrt(2), 1+(sqrt(2))/2

I think you misplaced a parenthesis, and that you meant to say

1 + sqrt(2), 1 + (sqrt(2)/2)

I admit that the operator precedence rules for most computer languages would perform the division operation before the addition operation anyway, but I think most people would agree that the meaning is more clear if the parentheses are arranged as shown in the second example.

Take Care, Barry

[Gerson W. Barbosa]

(05-10-2016 09:17 AM)BarryMead Wrote:  

(05-09-2016 08:29 PM)Gerson W. Barbosa Wrote:  1+sqrt(2), 1+(sqrt(2))/2

I think you misplaced a parenthesis, and that you meant to say

1 + sqrt(2), 1 + (sqrt(2)/2)

I admit that most machine code parsing algorithms would perform the division operation before the addition operation anyway, but I think most people would agree that the meaning is more clear if the parentheses
are arranged as shown in the second example.

Take Care, Barry

Actually only one pair of parentheses would have been enough: 1 + sqrt(2)/2. But redundant parentheses wouldn't hurt: (1+((sqrt(2))/2))

This is better: \(1+\frac { \sqrt { 2 } }{ 2 } \)

Gerson.

[Paul Dale]

Gerson,

Quote the reply and see how it is done. Then create your own response.

\(1+\frac 1 { \sqrt { 2 } } \)


Pauli

[Gerson W. Barbosa]

(05-10-2016 11:14 AM)Paul Dale Wrote:  Gerson,

Quote the reply and see how it is done. Then create your own response.

\(1+\frac 1 { \sqrt { 2 } } \)


Pauli

Thanks! I'd just figured that out and edited my reply before I saw your post.

This used to work in the old forum and still does :-)

Gerson.

[Gerson W. Barbosa]

(05-09-2016 10:00 PM)Tugdual Wrote:  I also have \(1-\frac { \sqrt { 2 } }{ 2 } \) and \(1-\sqrt { 2 } \)
Thanks 50G

Ooops sorry it was said positive numbers. Only one solution then.

I didn't take a 50g for this one, but I did use the wp34s to solve a quadratic equation:

1 ENTER 2 +/- 1 +/- SLVQ --> 2.41421356237 x<>y --> -.414213562373⁻¹

Gerson.

[David Hayden]

Thanks for playing folks. It's interesting to graph to two functions also. You'll see that there is a limit to the solution at (0,0). Hmm. I'm sure I have the terminology wrong there, but the x+y=x*y has a solution at (0,0) and x-y-y/x = epsilon as x,y approach 0 (for the right small values of x&y)

[Gerson W. Barbosa]

(05-10-2016 03:08 PM)David Hayden Wrote:  Thanks for playing folks. It's interesting to graph to two functions also. You'll see that there is a limit to the solution at (0,0). Hmm. I'm sure I have the terminology wrong there, but the x+y=x*y has a solution at (0,0) and x-y-y/x = epsilon as x,y approach 0 (for the right small values of x&y)

Nice little problem. Thanks for posting! It reminds me of another one, worked-out allegedly in half a minute by a 14-year old student, about one hundred years ago:

\(\left \{ _{\sqrt{y}+x=11}^{\sqrt{x}+y=7} \right.\)

No ordinary student, though.

Gerson.

[damaltor]

These are too easy if you sit in front of a computer.

[Gerson W. Barbosa]

(05-10-2016 05:22 PM)damaltor Wrote:  These are too easy if you sit in front of a computer.

Neither computer nor calculator needed, at least for the previous one:

[Tugdual]

(05-10-2016 05:07 PM)Gerson W. Barbosa Wrote:  

(05-10-2016 03:08 PM)David Hayden Wrote:  Thanks for playing folks. It's interesting to graph to two functions also. You'll see that there is a limit to the solution at (0,0). Hmm. I'm sure I have the terminology wrong there, but the x+y=x*y has a solution at (0,0) and x-y-y/x = epsilon as x,y approach 0 (for the right small values of x&y)

Nice little problem. Thanks for posting! It reminds me of another one, worked-out allegedly in half a minute by a 14-year old student, about one hundred years ago:

\(\left \{ _{\sqrt{y}+x=11}^{\sqrt{x}+y=7} \right.\)

No ordinary student, though.

Gerson.

I could NUM.SLV it but no chance with CAS on the 50g, keeps saying "not exact system" whatever this means...

[Gerson W. Barbosa]

(05-10-2016 06:58 PM)Tugdual Wrote:  I could NUM.SLV it but no chance with CAS on the 50g, keeps saying "not exact system" whatever this means...

What are your initial guesses? Starting with [1. 1.] the answers are found in 23 seconds. [0. 0.] gives me "- Error: Negative Underflow".

Gerson.

[Massimo Gnerucci]

(05-10-2016 06:58 PM)Tugdual Wrote:  I could NUM.SLV it but no chance with CAS on the 50g, keeps saying "not exact system" whatever this means...

Do you really need a calc for this one?

[Tugdual]

(05-10-2016 08:22 PM)Gerson W. Barbosa Wrote:  

(05-10-2016 06:58 PM)Tugdual Wrote:  I could NUM.SLV it but no chance with CAS on the 50g, keeps saying "not exact system" whatever this means...

What are your initial guesses? Starting with [1. 1.] the answers are found in 23 seconds. [0. 0.] gives me "- Error: Negative Underflow".

Gerson.

I used [5 5] and it does work for NUM.SLV but the error is returned with the standard SOLVE.

(05-10-2016 08:25 PM)Massimo Gnerucci Wrote:  Do you really need a calc for this one?

Why not? This is kind of a cal forum here...

[Massimo Gnerucci]

(05-10-2016 09:13 PM)Tugdual Wrote:  

(05-10-2016 08:25 PM)Massimo Gnerucci Wrote:  Do you really need a calc for this one?

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=?

:)

[Paul Dale]

(05-10-2016 05:07 PM)Gerson W. Barbosa Wrote:  \(\left \{ _{\sqrt{y}+x=11}^{\sqrt{x}+y=7} \right.\)

I got the answer without a calculation aid in well under thirty seconds.
I'm several times 14 though.


Pauli

[Dieter]

(05-10-2016 10:45 PM)Paul Dale Wrote:  I got the answer without a calculation aid in well under thirty seconds.
I'm several times 14 though.

How did you do it?

I have to admit I only got a (very) quick-and-dirty solution: Since x=(7–y)² and y=(11–x)² I assumed x and y to be the squares of integers, i.e. 4 or 9 or 16, etc. A simple substitution yields 22x – x² – sqrt x = 114. Omitting the root as the smallest term yields x = 11 ± sqrt 7, i.e 8,35 and 13,65. So two candidates are 9 and 16. I first tried x=9, et voilà... #-)

Dieter

[Paul Dale]

(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