Dear Tim,
Hi. We deleted this mesage about 5 times but here it is. We found a pattern. You have your basic numbers 1-12 using only the letters A & P. then, the numbers 13-24 you start using the letter E .At twenty-five you do E+1=P=EP=26 and 27=E+2=PP=EPP and so on until 63. At 63 You take 125 - 63 that equals 62 so you do Geeaapp. This is 125 - 63=62 the eeaapp is -62. After 63 for 64 you have 125-64=63 or Geeaap and so on until you reach 125 which is just G. Then for 126 you do G+1=P=GP=126. For 127 you do G+2=PP=GPP=127. You follow this pattern as we explained up to 250. Two fifty is GG and 251 is GG+1=P=GGP=251 . You follow this pattern and you find that you cannot do 313. From Jen, Tim, Leah, Andrea & Robby

Megan, Jennifer, Ashley, Joslynn, and Matthew of Monroe City, MO thought that this problem, GEAP!, was a little tougher than Znorlian. It took us about thirty minutes to complete this problem. Here is what we came up with:

					G= 125   g= -125
					E=  25   e=  -25
					A=   5   a=   -5
					P=   1   p=   -1
					

The first clue told us A= 5, AA= 10. The second clue told us PA= 6 and pA= 4. And since A was 5, P had to be 1 and p=-1 because clue third says lower case letters are negative and upper case letters are positive. The third clue also says GAA= 135, AA= 10, so G= 125. It also says no letter can be used more than twice in a number. Clue four says it doesn't matter what order the letters are in and that Ep= 24, since p= -1, E has to equal 25. The only way to make 13 is AApAp, but you can't do that because A is used more than twice. Here is a list of numbers up to 12:

1= P
2= PP
3= pAp
4= pA
5= A
6= AP
7= APP
8= pApA
9= AAp
10=AA
11=AAP
12=AAPP

teacher: Mrs. Barbara Carson, Monroe City Middle School [1997, I think]

Rachel Jones: I believe that every number can be made until you run out of letters. You can use each letter twice, so you would put G,E,A,and P twice. Then 1 number above that is impossible to make correctly in the geap method. All the letters put twice is equal to 312. So 312 + 1 = 313--Impossible: if A=5, a= -5, P=1, p= -1, E=25, so e= -25. John Strader: We found that you can make up the biggest number possible GGEEAAPP = 312 because if there is a number that's a multiple of 5, there is no P needed. You can make every multiple of 5 up to 310 using geap. You can make a number ending in 3 or 4 by making number end in 5 and add a certain number of little p's. For a number ending in 1 or 2, make a number ending in 0 then add a certain number of big P's. The highest is 312 so the next one you can't get is 313. You can't go bigger than 312 because you are only allowed to use 2 of the same Geap letters. (example: Geapp is fine; however Geappp is not fine).

Note from the teacher: My students noticed immediately that they were unable to make 313 in GEAP. The difficult part was when I asked how they knew this was the smallest number they couldn't make. They stuggled for several class periods with this. Here is their final explanation. -Karen Jagolinzer, Harrison Middle School, Yarmouth, ME.

G-E-A-P (From Mrs. Jagolinzer's 7th Grade Math Think Lab Class)

G (g) = 125 E (e) = 25 A (a) = 5 P (p) = 1

Two numbers next to each other is addition. Capitalized letters are positive numbers and lowercase letters are negative. Therefore, "AA" is adding together two positive numbers, "aa" is adding together two negative numbers, "Aa" is adding a negative number to a positive number, and "aA" is adding a positive number to a negative number. It is a given that "A" = 5. Since "PA" = 6, "P" has to be 1. And also, if pA = 4, then "p" has to be -1. If "Ep" = 24, knowing that "p" = -1, "E" has to be 25. Since "PEG" = 151, and "PE" = 26, "G" has to be 125.

We found that we could make the numbers 1-12 using just "P's" and "A's". Then, if you add an "E", you can get any number between, 25-12 and 25+12 using the "P's" and "A's" (positive and negative), because "E" is 25 and if you can get all of the numbers 1-12 using "A's" and "P's" than you can make 25 + (#'s 1-12) using just "A's", "P's", and "E". By adding another "E" you can get the numbers 50 - (#'s 1-12) and 50 +(#'s 1-12). From there, you can add a "G" to get "G"-( #'s 1-62) and "G"+(#'s 1-62). This gives you all of the numbers from 1 to 162. By adding another "G" you can get any number up to 312. If you put all of the numbers together (in positive form, "GGEEAAPP") the highest number you can is 312 and if you can get all of the numbers from 1-312, but nothing higher, the lowest number you can't make is 313.

To find the answer we first found that the highest number you can make on GEAP is GGEEAAPP (312), because you may only use two of each letter. Then we found how to make the numbers 1-12. P=1, PP=2, App=3, Ap=4, A=5, AP=6, APP=7, AApp=8, Aap=9, AA=10, AAP=11, AAPP=12, After trying several strategies we came up with a chart that by using the numbers 1-12 we could prove what the lowest number you couldn't make on GEAP is.

We had already made 12 so basically itās 1-12 is the highest you can go with just A and P so we added E (25). Then E(25)aapp(-12)=13 and if we can make 1-12 then we can make 13-25 and if we do uppercase instead of lowercase then that gives us the numbers through 37, then we added the other E so itās EEaapp to get us 38 then we do the same thing we did with the first E set bring us up to 62(EEAAPP). That is the highest number we are able to make with E, A, P so we then added the first G(125). We then repeated the process like what we did with Eās and 12 only we did it with 62 giving us the numbers 1-187. Finally we added the last G and we did 188(Ggeeaapp)-312(GGEEAAPP).

					312 GG+62
					250 GG+62188 GG-62
					187 G+62
					125 G
					 63 G-62
					 62 EE+12
					 38 EE-12
					 37 E+12
					 25 E
					 13 E-12
					

So we are saying the lowest, positive, number you can't make on GEAP is 313 because you can only make the numbers 1-312.

Annie, Jon, and Adam