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

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.




Procrustes -> RE: Math Poser (1/15/2007 8:49:48 PM)

#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.




sterckxe -> RE: Math Poser (1/15/2007 9:17:20 PM)

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) [8D]

Greetz,

Eddy Sterckx




Shannon V. OKeets -> RE: Math Poser (1/15/2007 9:37:20 PM)

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) [8D]

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.




JAMiAM -> RE: Math Poser (1/16/2007 1:18:20 AM)

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.




Shannon V. OKeets -> RE: Math Poser (1/16/2007 2:06:46 AM)

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.




Laryngoscope -> RE: Math Poser (1/16/2007 2:28:55 AM)

Ok for the easiest

29 = 18

edit:

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




Laryngoscope -> RE: Math Poser (1/16/2007 2:39:10 AM)

Cheeky solution [:D]

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");
}




Shannon V. OKeets -> RE: Math Poser (1/16/2007 3:26:39 AM)


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).




epistibrain -> RE: Math Poser (1/16/2007 5:05:57 AM)

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.




Procrustes -> RE: Math Poser (1/16/2007 5:33:54 AM)

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




Shannon V. OKeets -> RE: Math Poser (1/16/2007 5:54:02 AM)


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. [&o] 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 ...[:D]




SemperAugustus -> RE: Math Poser (1/16/2007 4:01:56 PM)

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