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Math Poser - 1/15/2007 8:15:16 PM   
Shannon V. OKeets

 

Posts: 15464
Joined: 5/19/2005
From: Honolulu, Hawaii
Status: online 		A friend of mine posed a math problem to me last month. Here is a modified form of that problem/poser; I’ve made it both easier and more difficult.

Given the math statement:
21 = 98
manipulate the 5 symbols so they make a true math statement. The given statement is clearly incorrect. No additional symbols can be added, nor can any of the 5 symbols be removed.

There are 3 conceptual solutions to this problem, and within each of the solutions, several variations are possible.

If you want some hints, continue reading.







HINTS

The easiest solution involves moving only 2 symbols.





The solution of moderate difficulty involves moving 1 symbol.





The hardest solution involves moving no symbols.

_____________________________

Steve
Post #: 1
RE: Math Poser - 1/15/2007 8:49:48 PM   
Procrustes

 

Posts: 633
Joined: 3/30/2003
From: Upstate
Status: offline 		#1: 9^2 = 81 (nine squared equal eighty-one)

(or perhaps you meant 9(2) = 18? Though then I have to add the muliplication operator....)

The other's aren't so readily apparent to me - good puzzle.


< Message edited by Procrustes -- 1/15/2007 9:03:51 PM >

(in reply to Shannon V. OKeets)
Post #: 2
RE: Math Poser - 1/15/2007 9:17:20 PM   
sterckxe


Posts: 4593
Joined: 3/30/2004
From: Flanders
Status: offline
quote:

ORIGINAL: Shannon V. OKeets

Given the math statement:
21 = 98
manipulate the 5 symbols so they make a true math statement. The given statement is clearly incorrect. No additional symbols can be added, nor can any of the 5 symbols be removed.

The hardest solution involves moving no symbols.


Easy : 98 is in base 11 (107 decimal) and 21 in base 53 (107 decimal)

Greetz,

Eddy Sterckx

(in reply to Shannon V. OKeets)
Post #: 3
RE: Math Poser - 1/15/2007 9:37:20 PM   
Shannon V. OKeets

 

Posts: 15464
Joined: 5/19/2005
From: Honolulu, Hawaii
Status: online
quote:

ORIGINAL: sterckxe
quote:

ORIGINAL: Shannon V. OKeets

Given the math statement:
21 = 98
manipulate the 5 symbols so they make a true math statement. The given statement is clearly incorrect. No additional symbols can be added, nor can any of the 5 symbols be removed.

The hardest solution involves moving no symbols.


Easy : 98 is in base 11 (107 decimal) and 21 in base 53 (107 decimal)

Greetz,

Eddy Sterckx

Nice, but then wouldn't you need to add notation for the different bases? The solution I have in mind is different.

_____________________________

Steve

(in reply to sterckxe)
Post #: 4
RE: Math Poser - 1/16/2007 1:18:20 AM   
JAMiAM

 

Posts: 6146
Joined: 2/8/2004
Status: offline 		The problem is poorly posed...

Define "manipulate" with regard to a symbol. Define "move".

Are mathematical operations allowed to be implied?

My own submission is as follows:

Allow each of the individual numbers to be mapped as both the x,y coordinates in Quadrant I of the standard (Cartesian) plane. That is, 2 is implied to equal the point (2,2). The number 1 is implied to equal the point (1,1), 9 equals point (9,9), and 8 equals point (8,8). Finally, define a two digit sequence "ab" to be the drawing of the line segment between the two points a and b. Thus, 21 is the line segment containing points (2,2) and (1,1), while 98 is the line segment containing the points (9,9) and (8,8).

Then, when we compare the lines defined by the two above described line segments, we see that they are indeed, both the same.

There...no symbols "moved" and only "manipulated" in the sense of being used within defining statements, and mathematical operations.

(in reply to Shannon V. OKeets)
Post #: 5
RE: Math Poser - 1/16/2007 2:06:46 AM   
Shannon V. OKeets

 

Posts: 15464
Joined: 5/19/2005
From: Honolulu, Hawaii
Status: online
quote:

ORIGINAL: JAMiAM

The problem is poorly posed...

Define "manipulate" with regard to a symbol. Define "move".

Are mathematical operations allowed to be implied?

My own submission is as follows:

Allow each of the individual numbers to be mapped as both the x,y coordinates in Quadrant I of the standard (Cartesian) plane. That is, 2 is implied to equal the point (2,2). The number 1 is implied to equal the point (1,1), 9 equals point (9,9), and 8 equals point (8,8). Finally, define a two digit sequence "ab" to be the drawing of the line segment between the two points a and b. Thus, 21 is the line segment containing points (2,2) and (1,1), while 98 is the line segment containing the points (9,9) and (8,8).

Then, when we compare the lines defined by the two above described line segments, we see that they are indeed, both the same.

There...no symbols "moved" and only "manipulated" in the sense of being used within defining statements, and mathematical operations.

The problem is intentionally loosely stated, to allow the imagination to roam. Your solution could be reduced to a single dimension, with the distance from 1 to 2 being the same as the distance from 8 to 9. I like the visual aspects of your solution, but it does have that assumption about an underlying grid.

This is not the third conceptual solution I have. Actually, no one has even given the second solution so far.

_____________________________

Steve

(in reply to JAMiAM)
Post #: 6
RE: Math Poser - 1/16/2007 2:28:55 AM   
Laryngoscope


Posts: 1085
Joined: 7/8/2004
From: Newcastle, Australia
Status: offline 		Ok for the easiest

29 = 18

edit:

doh ... as above (first reply), multiplication is implied as in XY = Z


< Message edited by Laryngoscope -- 1/16/2007 2:43:35 AM >

_____________________________

"It is not a Commonwealth division, it is an Australian Division. Why, give me two Australian Divisions and I will conquer the world for you!"

-- Erwin Rommel

(in reply to Shannon V. OKeets)
Post #: 7
RE: Math Poser - 1/16/2007 2:39:10 AM   
Laryngoscope


Posts: 1085
Joined: 7/8/2004
From: Newcastle, Australia
Status: offline 		Cheeky solution

Manipulate the "=" sign, crack it in half to get two "-" then

21 - 9 - 8 (which = 5) which evals as true in most program language statements 

if (21 - 9 - 8)
{
   printf("It's true!");
}


Or just rotate the "=" sign as in

21 || 98

which is also true 

if (21 || 98)
{
   printf("This is true too");
}


< Message edited by Laryngoscope -- 1/16/2007 2:56:42 AM >

_____________________________

"It is not a Commonwealth division, it is an Australian Division. Why, give me two Australian Divisions and I will conquer the world for you!"

-- Erwin Rommel

(in reply to Laryngoscope)
Post #: 8
RE: Math Poser - 1/16/2007 3:26:39 AM   
Shannon V. OKeets

 

Posts: 15464
Joined: 5/19/2005
From: Honolulu, Hawaii
Status: online
quote:

ORIGINAL: Laryngoscope

Ok for the easiest

29 = 18

edit:

doh ... as above (first reply), multiplication is implied as in XY = Z


Not to my liking.

For the first solution I prefer the exponential solution given by the 1st responder of 9 raised to the 2nd power equals 81. Positioning the 2 above and to the right of the 9 is standard for denoting an exponent. Seeing a multiplication when no symbol is shown is okay for letters (algebra) but not for numbers (arithmetic).

_____________________________

Steve

(in reply to Laryngoscope)
Post #: 9
RE: Math Poser - 1/16/2007 5:05:57 AM   
epistibrain

 

Posts: 1
Joined: 1/16/2007
Status: offline 		For a solution moving two symbols, you could rearrange the 1 and the 9 to get 2 = 8 - 6.

For a solution moving just one symbol you could move the 1 to negate the equal sign, to get 2 not equal to 98.

Of course, both of these look a little more reasonable when handwritten than in most fonts, so that a horizontal 1 is a credible minus sign.

I can't think of any way to do it without moving any symbols, but I would be curious to hear a solution.

(in reply to Shannon V. OKeets)
Post #: 10
RE: Math Poser - 1/16/2007 5:33:54 AM   
Procrustes

 

Posts: 633
Joined: 3/30/2003
From: Upstate
Status: offline
quote:

ORIGINAL: Shannon V. OKeets

21 = 98

The solution of moderate difficulty involves moving 1 symbol.


I'm having a hard time deciding what to deam "moving a symbol", but here's another (where one digit is copied...)

1/2 = 9/18



EDIT - typo

< Message edited by Procrustes -- 1/16/2007 5:50:02 AM >

(in reply to Shannon V. OKeets)
Post #: 11
RE: Math Poser - 1/16/2007 5:54:02 AM   
Shannon V. OKeets

 

Posts: 15464
Joined: 5/19/2005
From: Honolulu, Hawaii
Status: online
quote:

ORIGINAL: epistibrain

For a solution moving two symbols, you could rearrange the 1 and the 9 to get 2 = 8 - 6.

For a solution moving just one symbol you could move the 1 to negate the equal sign, to get 2 not equal to 98.

Of course, both of these look a little more reasonable when handwritten than in most fonts, so that a horizontal 1 is a credible minus sign.

I can't think of any way to do it without moving any symbols, but I would be curious to hear a solution.


Excellent for the 1st one. I had missed that, though I did purposely include a 9 so it could be used as a 6. [I had the solution 9 squared = 81.]

And your solution for moving the 1 to make the not-equal sign is what I had as the moderate difficulty solution.

For the third, I have (please don't everyone hate me now), print the line 21 = 98 out and cut it from a piece of paper; then tape the left edge to the right edge so it reads (as you turn it around) 9821 = 9821 = 9821 ...

_____________________________

Steve

(in reply to epistibrain)
Post #: 12
RE: Math Poser - 1/16/2007 4:01:56 PM   
SemperAugustus

 

Posts: 257
Joined: 1/9/2005
Status: offline 		<Deleted>

< Message edited by SemperAugustus -- 1/16/2007 4:14:02 PM >

(in reply to Shannon V. OKeets)
Post #: 13
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