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Subject: "pentagon addition"     Previous Topic | Next Topic
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Conferences The CTK Exchange Early math Topic #31
Reading Topic #31
Victorious-Not
guest
Nov-10-01, 11:13 PM (EST)
 
"pentagon addition"
 
   Using numbers 1-10 place the numerals on each corner with one numeral in the midline. use each numeral 1 time, make the 3 numerals on each side add to the same answer, there is a possible 4 different sums. Remember each side of the pentagon has the same sum. I cant even solve it for one sum on all sides.


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  Subject     Author     Message Date     ID  
  RE: pentagon addition Hirdesh Nov-12-01 1
     RE: pentagon addition DoubleE Jan-07-04 2
         RE: pentagon addition alexbadmin Jan-07-04 3
             RE: pentagon addition DoubleE Jan-07-04 4
                 RE: pentagon addition alexbadmin Jan-08-04 5
                     RE: pentagon addition DoubleE Jan-08-04 6
                         RE: pentagon addition alexbadmin Jan-08-04 7
                             RE: pentagon addition DoubleE Jan-08-04 8
                                 RE: pentagon addition alexbadmin Jan-08-04 9
                                     RE: pentagon addition DoubleE Jan-08-04 10
                                         RE: pentagon addition alexbadmin Jan-08-04 11

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Hirdesh
guest
Nov-12-01, 12:17 PM (EST)
 
1. "RE: pentagon addition"
In response to message #0
 
   let the sum of each side be alpha

now alpha can be either odd or even.

as all the numbers on all sides cannot be all even or odd therefore either it is odd+odd+even or even+even+odd. thus the numbers on the middle of segments are either all odd or even.

further 55+sum of coners = 5 * alpha.

thus alpha = 16 or 17.

for 16 ( even in the middle ) 2 combinations has to be 5-10-1 and 9-2-5.
thus one solution is

5-2-9-4-3-6-7-8-1-10-5.



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DoubleE
Member since Jan-7-04
Jan-07-04, 06:00 PM (EST)
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2. "RE: pentagon addition"
In response to message #1
 
   I need assistance with solving these types of problems. My first grader is bringing home several of these math "challenge" problesm each week. Some are pentagons, squares, H's, circles (like a ferris wheel), etc. The latest one is a hexagon, using the numbers 1 - 12 only once, with all six sides of the hexagon resulting in the same sum, using three numbers added together.

Is there an easy way to help with these problesm and make them fun, rather than frustrating for my child? I am not a math expert, so if anyone has some advice, please explain it in layman's terms. Thanks for any help you can provide!!!!


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alexbadmin
Charter Member
1169 posts
Jan-07-04, 06:13 PM (EST)
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3. "RE: pentagon addition"
In response to message #2
 
   Let's do that in two steps. First I need some information from you.


  1. You posted your message as a reply to another one. Have you read it? If not, please do it now. Let me know what you think.


  2. What was the previous problem? How did you manage to solve it? If you did not, have you discussed the solution with your kid after the work was graded?

    Thank you


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DoubleE
guest
Jan-07-04, 09:20 AM (EST)
 
4. "RE: pentagon addition"
In response to message #3
 
   Hi! I did read the previous message regarding the pentagon problem, which we were able to solve through trial and error. I did try to apply your formula to the hexagon problem, with 78 being the sum of all the numbers, but frankly, I got lost after that. I'm a Psychologist by trade, so all of this is a bit foreign to me.

The previous 10 to 15 problems we have solved through trial and error or simply by luck. There is no grading of these problems - these are just challenges for children (like my son) who love math!

We have been working on this problem for may days and my son is very frustrated because we cannot solve it. We're currently working on the assumption that the sum is 19 (having tried 16, 17, 18), but have yet to be successful.

Again, I'm looking for a little help on how to approach these problems so they can be "fun" rather than "frustrating". I appreciate any advice you can provide!!!


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alexbadmin
Charter Member
1169 posts
Jan-08-04, 10:01 AM (EST)
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5. "RE: pentagon addition"
In response to message #4
 
   >Hi! I did read the previous message regarding the pentagon
>problem, which we were able to solve through trial and
>error. I did try to apply your formula to the hexagon
>problem, with 78 being the sum of all the numbers, but
>frankly, I got lost after that. I'm a Psychologist by
>trade, so all of this is a bit foreign to me.

As with the pentagon, there are two possibilities. The numbers on every side form one of two possible patterns: it's either odd-even-odd or even-odd-even.

The sum of all numbers 1 through 12 is 78. That of the odd ones is 36. The even numbers add up to 42. Therefore, for the pattern o-e-o the sum on a side must be (78 + 36)/6 = 19. (This is because the numbers at the corners are counted twice, while the numbers at midpoints are counted only once.) But this is impossible, since the sum in the pattern o-e-o is bound to be even.

For the pattern e-o-e, the sum on a side must be (78 + 42)/6 = 20. This is also impossible because the sum in the pattern e-o-e is always odd.

>
>The previous 10 to 15 problems we have solved through trial
>and error or simply by luck. There is no grading of these
>problems - these are just challenges for children (like my
>son) who love math!

As a parent, I'd talk to the teacher. It does not matter whether the problems are graded or not. The teacher is there to facilitate the learning. Therefore, I would expect that some time is set aside to discuss all the assigned problems, whether graded or not. Especially so in the first grade. Children had to learn something from what they are doing and can't be counted upon to do that all on their own. I would complain to the principal, if the teacher refuse to discuss the problems in class.

>We have been working on this problem for may days and my son
>is very frustrated because we cannot solve it. We're
>currently working on the assumption that the sum is 19
>(having tried 16, 17, 18), but have yet to be successful.

As you have seen above, the problem has no solution.

>Again, I'm looking for a little help on how to approach
>these problems so they can be "fun" rather than
>"frustrating". I appreciate any advice you can provide!!!

All the problems of that sort are solved in the manner outlined above. Some numbers may appear more than others. What may those be? You always must be able to determine the possible common sum up front. Once you do, this sum could be split only in so many ways without repetitions. List all them down. Try to place the sums from the list on the side of the given shape. This is always a combination of a little theory and trial-and-error. Never start trial-and-error before you found the common sum.


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DoubleE
guest
Jan-08-04, 03:15 PM (EST)
 
6. "RE: pentagon addition"
In response to message #5
 
   Thanks for your help! We were doing as you suggested by determining a sum and then listing all possible combinations of number (this example, using three numbers) along the side of the figure and then placing them in the figure through trial and error. We did try previous sums (16, 17, 18) and then one child in the class solved the problem using the sum of 19, so we gave that sum a try. How is it possible that the child could solve it using 19? We can get it to balance on all sides but one, and you stated that it is not possible to solve.

I will certainly speak to his teacher as this process is becoming increasingly more frustrating. These challenge problems are given in a series of three (for examplee, we are on level E). Once the child solves all three problems in the level, he/she is given a new challenge book. My son is very competetive and is frustrated someone else in the class has moved ahead of him. According to his teacher, all of these problems can be solved.

These challenge books appear to be reproduced from some sort of book. Are you, or anyone else familiar with a book like this?

We'll keep plugging away! Again, thanks for your assistance!


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alexbadmin
Charter Member
1169 posts
Jan-08-04, 04:28 PM (EST)
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7. "RE: pentagon addition"
In response to message #6
 
   >We did try previous sums (16, 17, 18) and
>then one child in the class solved the problem using the sum
>of 19, so we gave that sum a try. How is it possible that
>the child could solve it using 19?

Well, if a first grader did this I should be able to. So I went and checked what I wrote. Indeed other patterns are possible. Assume the common sum is odd, say, like 19. From what I wrote it follows that we can't get that with the pattern odd-even-odd on each side, nor with the pattern even-odd-even. OK. We can get an odd number as either the sum odd+odd+odd or odd+even+even. The sides then should be

o+o+o-o+e+e-e+e+o-o+o+o-o+e+e-e+e+o-,

where o-o and e-e stand for a single number in a corner, while the three numbers joined by "+" lie on a single side. Thus the must be 2 sides with o+o+o pattern and 4 sides with o+e+e pattern.

How can 19 be represented as the sum o+o+o with the largest odd summand being 11 and all three being different?

19 = 11 + 1 + 7
19 = 11 + 3 + 5
19 = 9 + 1 + 9 <- can't be used, because of two 9's
19 = 9 + 3 + 7
19 = 9 + 5 + 5 <- can't be used
19 = 7 + 5 + 7 <- can't be used

So, in fact, there are three possibilities to have the o+o+o pattern:
11+1+7, 11+3+5, 9+3+7. We should choose two non-overlapping pairs. But we can't. For example, 11+1+7 and 9+3+7 overlap at 7. Aha. So, if the problem is solvable, the pattern must allow for two contiguous o+o+o sides:

o+o+o-o+o+o-o+e+e-e+e+o-o+e+e-e+e+o-

Now we need to represent 19 as the sum e+e+o (we'll need four such):

19 = 12 + 1 + 6
19 = 12 + 3 + 4
19 = 12 + 5 + 2
19 = 10 + 1 + 8
19 = 10 + 3 + 6
19 = 10 + 5 + 4
19 = 10 + 7 + 2
19 = 8 + 5 + 6
19 = 8 + 7 + 4
19 = 8 + 7 + 2
19 = 6 + 9 + 4
19 = 6 + 11 + 2

So we are looking for the pattern

o +o+o-o+o+o-o+ e+e-e+ e+o-o+e+e-e+e+o-

11+1+7-7+9+3-3+12+4-4+10+5-5+8+6-6+2+11-

Here I used "trial-and-error" and some common sense which I'd have hard time sharing. So, put 11 in a corner and continue in this way:

11, 1, 7, 9, 3, 12, 4, 10, 5, 8, 6, 2

>We can get it to balance
>on all sides but one, and you stated that it is not possible
>to solve.

Well, do not get deluded into thinking that if only one side remains, the problem is solvable.

>I will certainly speak to his teacher as this process is
>becoming increasingly more frustrating. These challenge
>problems are given in a series of three (for examplee, we
>are on level E). Once the child solves all three problems
>in the level, he/she is given a new challenge book.

I do not understand how is it possible that solutions are not being discussed.

>My son
>is very competetive and is frustrated someone else in the
>class has moved ahead of him.

This kind of problems are both difficult and simple. Trial-and-error alone should not be used. Luck will fail you sooner or later.

>According to his teacher, all
>of these problems can be solved.

OK.

>These challenge books appear to be reproduced from some sort
>of book. Are you, or anyone else familiar with a book like
>this?

I am aware of a book by Peggy Kaye Games for Math (from Kindergarten to 3rd grade) where similar problems go under the caption of Number Bubbles, where numbers are put into "bubbles" - circles - and the bubbles are arranged as crosses, stars, triangles. etc. But all the puzzles in the book are small, nowhere close to using 12 numbers.

>We'll keep plugging away!

So, again. One way or another try thinking first and use trial-and-error as the last resort (you'll always come to that any way.)


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DoubleE
guest
Jan-08-04, 06:19 PM (EST)
 
8. "RE: pentagon addition"
In response to message #7
 
   WOW! You put way too much effort into your response! Believe it or not, we were able to solve it just a short ime ago using the following : 1,10,8,2,9,6,4,12,3,5,11,7,1. I keep checking it again and again, but I think it works!

I appreciate all of your help and will use all I have learned with our next packet! I'm sure I'll be back begging for help in no time. Many, many thanks!!!!!


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alexbadmin
Charter Member
1169 posts
Jan-08-04, 06:29 PM (EST)
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9. "RE: pentagon addition"
In response to message #8
 
   >WOW! You put way too much effort into your response!
>Believe it or not, we were able to solve it just a short ime
>ago using the following : 1,10,8,2,9,6,4,12,3,5,11,7,1. I
>keep checking it again and again, but I think it works!

Yap, it does. It is different from mine, too. Hard work pays.

But now a really good advice: think of your solution, put it in some framework, do not leave it just "trial-and-error", if it were of course. For example, your solution fits the same pattern

o+o+o-o+o+o-o+e+e-e+e+o-o+e+e-e+e+o-

(this is if you rotate your pattern to start with 11.)

My original patterns made up of e-o-e and o-e-o did not work. (An I was mistaken to think there were no other patterns.) It would be very useful to your boy to have a look back and try to understand why one pattern worked while others did not.


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DoubleE
guest
Jan-08-04, 09:13 PM (EST)
 
10. "RE: pentagon addition"
In response to message #9
 
   Again, many,many thanks for your help! We are so fortunate to have found this site! If you are at all interested, we would be happy to e-mail to you the previous puzzles we have solved. Perhaps you would be interested in posting them on your site - perhaps others would find these fun (and frustrating!) as well!


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alexbadmin
Charter Member
1169 posts
Jan-08-04, 09:16 PM (EST)
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11. "RE: pentagon addition"
In response to message #10
 
   >Again, many,many thanks for your help! We are so fortunate
>to have found this site!

You are welcome. No need for this profusion.

>If you are at all interested, we
>would be happy to e-mail to you the previous puzzles we have
>solved. Perhaps you would be interested in posting them on
>your site - perhaps others would find these fun (and
>frustrating!) as well!

By all means. Post them here. Let people see what American firstgraders are doing nowadays. Thank you.


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