THE TROOP RIDDLE ANSWSER: THE REAL SOLUTION
The troop riddle continues to confound people, read some of the plausible answers that have been suggested. Is one of these explanations the correct one?
Explanations of The Troop Riddle
Version 1 (as we presented it the first time)
Stopping along the trail to rest, Judy saw that one-third of the rest of the group from the Girl Scouts was behind her and three-fourths were ahead. Not counting Judy, what was the total number in her group? Answer: 13
Version 2 and the explanation for giving the answer 13:
Resting along a path, Judy noticed that she and those behind her made up 1/3 of the troop and that those ahead made up 3/4 of troop. Counting Judy, what would be the smallest number of hikers that day? The answer is 13. Judy looked behind her and sees 3, plus her
would be 1/3 (4 out of 12). Ahead she sees 9 or 3/4 of 12. The total counting Judy is 13. That would be the smallest number as any number divisible by both 3 and 4 could answer this riddle! This isn't the best math question, but remember it is a riddle and riddles are designed to "play on words" and misdirect the person reading them. In this riddle, you are dealing with the number in the troop plus Judy, but Judy includes herself to get the 1/3 figure. And the final question wants to include Judy in the total but she isn't in the total of the hikers ahead. Little twists and turns are what makes a math question into a math riddle.
Version 3
Stopping along the trail to rest, Judy saw that one-third of the rest of the group from the Girl Scouts was behind and three-fourths were ahead. Not counting Judy, what was the total number in her group?
Randy writes: The puzzle leaves the reader to make a lot of assumptions such as:
head of her, behind those that are ahead, etc. With the following definitions it becomes logical;
Persons can not be divided into fractions.
The group = 16. "The rest of the group" (not counting Judy) = 15, 1/3 of "the rest of the group" is behind (those that are ahead) = 5
Judy is part of the group. Those behind Judy = 4. 3/4 (of the group) were ahead (of those behind) =12
Those ahead of Judy =11
Not counting Judy, 11+4 = 15
And now for the REAL SOLUTION:
Your answer of 13 cannot be correct. In the first half of the riddle (she and those behind her), Judy includes herself in the total, and thus counts as part of the troop. In the second half (those ahead) she excludes herself from the troop total. Since the riddle includes her in the total, she is counted as part of the troop. She looks behind her, sees three people, and adds herself as the fourth. In doing this, she is including herself in the group. Four is not 1/3 of 13. She looks ahead and sees eight, which is indeed 3/4 of the REST of the group. So here are the variables: Total hikers: X -- This number INCLUDES Judy
She, and those behind: 1/3(X) -- This number also includes Judy
Those ahead: 3/4(X - 1) -- Remember, she isn't including herself in the group here!
Here is the math:
X = 1/3(X) + 3/4(X - 1) -- Total of group equals those in back plus those in front.
12X = 4X + 9(X - 1) -- Multiply both sides by 12
12X = 4X + 9X - 9 -- Factoring
12X = 13X - 9 -- Add similar variables
12X - 13X = -9 -- Subtract 13X from both sides
-X = -9 -- Add similar variables
X = 9 -- Multiply both sides by -1
So the answer is 9. This answer is not just the smallest number, it is the ONLY number, as the above math proves.
Does this fit? Here is the riddle once again: Resting along a path, Judy noticed that she and those behind her made up 1/3 of the troop and that those ahead made up 3/4 of troop. Counting Judy, what would be the smallest number of hikers that day?
Here is a diagram of the troop:
F F F F F F J B B
Judy noticed that she and those behind her (3 total) made 1/3 of the troop (of 9).
F F F F F F (J B B) -- 3 / 9 = 1/3
Those ahead (6) made of 3/4 of troop (of 8, not including her).
(F F F F F F) B B -- 6 / 8 = 3/4
Thanks Roger for clearing up the answer to this riddle!
We have enjoyed the "twists and turns" this riddle has taken and appreciate the input from Roger, Randy and the others who offered it. Thanks for challenging us to think!!
UPDATE!! UPDATE!! UPDATE!! UPDATE!!
A recent visitor to our site offers a different solution for this riddle. It follows .... what do you think???
John writes: I would definitely agree with the following statement from the solution Version 2. "This isn't the best math question, but remember it is a riddle and riddles are designed to "play on words" and misdirect the person reading them."
Now lets look at the original riddle without re-writing it. What does the rest of the group mean? From Judy's view point it would be the entire group minus herself. What does the section "and three-fourths were ahead" mean - does this mean 3/4 of the rest of the group or 3/4 of the entire group ? Lets look at the math from the perspective that the rest of the group means the entire group minus Judy herself and the section ahead of her is a fraction of the entire group, in this case 3/4.
Let x = the total Number in the group.
Then (x-1) is the rest of the group. The entire troop minus Judy herself.
So what is the equation? 1/3(x-1) + 3/4(x) = x
This means that 1/3 of the rest of the troop the people behind Judy and 3/4 of the entire group are ahead of her. We now add the two sections of the group to find out the entire number of people in the group. To solve the equation we will multiply both sides of the equation by 12 as it is evenly divisible by both 3 and 4.
12[1/3(x-1) + 3/4(x)] = 12[x]
12[(x/3-1/3) + (3x/4)] = 12[x]
12x/3-12/3+36x/4 =12x
4x-4+9x = 12x
4x+9x = 12x + 4
13x = 12x +4
13x-12x = 4
x = 4
The total Number of people in the troop that day was 4 and the answer to the riddle would be the total number in the troop minus Judy (4-1) or 3. Not counting Judy there were 3 people in the troop. This is a smaller number than any of the other solutions given.
And now, another way to look at this riddle from Nick: Neither of the solutions is right because the question has been worded wrong in all versions. Correct wording:
Stopping along the trail to rest, Judy noticed that she and those behind her made up one-third of the troop and that those ahead made up three-fourths of the rest of the troop. How many people in total are in the troop? (or how many are in the troop excluding Judy, either will do).
Answers to this is 9
Her plus two behind is three. 3 is one third of nine. Important to notice the words "the troop," which includes Judy. Six others ahead 6 is three-fourths of 8. Important to notice the words "rest of troop," which excludes Judy.
The other versions don't work because they refer to either both times the entire troop, or both times the rest of the troop, and not one of each. So the answer would be 4, in the troop, if the wording was the other way around, e.g. one-third of rest of troop and three-fourths the entire troop. So both randy and John had the right answer if the riddle was reworded correctly either way.
UPDATE AGAIN!!!
This riddle keeps getting more answers and here are two of the most recent: From Mike - Answer: 0
Reasoning: Riddles are a play on words, and if you take the literal meaning of the entire sentence, it never said she was a member of the Girl Scout Troop or that they were part of her group. If you take, "Not counting Judy", from her total group number of 1 (herself because no one else is explicitly mentioned as part of her group) then you have 0.
Version 3 has the same answer, but why reword the riddle to get a satisfactory answer. Just answer the original and stick to it. From Sau Fan Lee-
There are numerous (and different) solutions to your puzzle, but most of them are either incomplete or wrong. In this email, I'm going to analyze this puzzle logically and in GREAT details..
Let's start with Version 1. I'll also add in the some more information in brackets for the ambiguous parts of the puzzle:-
Stopping along the trail to rest, Judy saw that one-third of the rest of the group (i.e.. not counting herself) from the Girl Scouts was behind her and three-fourths (of what?) were ahead (of whom?). Not counting Judy, what was the total number in her group?
The first inserted brackets above just clarify that "the REST of the group" means Judy is not counted in the number.
The second inserted brackets shows the first ambiguity. Is it referring to "3/4 of the entire group (including Judy)" or "3/4 of the rest of the group (excluding Judy)"? Let's call this ambiguity: [A].
The third inserted brackets shows the second ambiguity. Is it referring to "ahead of Judy" or "ahead of those behind Judy"? Let's call this ambiguity: [B].
Now, the 4 possible solutions:-
1) Assume: [A] = the entire group (includes Judy) [B] = ahead of Judy x = the number of people in the ENTIRE group (includes Judy)
Then: Troops behind Judy = (x-1)/3
Troops in front of Judy = (x-1) * 2/3 (remember: Judy is not counted)
3/4 of ENTIRE group were ahead of Judy ==> x * 3/4
(x-1) * 2/3 = x * 3/4
(2x-2)/3 = 3x/4
8x-8 = 9x (multiply both sides by 12)
x = -8
As the number of people cannot be negative, we know this answer is impossible. So the initial assumption is wrong.
2) Assume: [A] = the entire group (includes Judy)
[B] = ahead of those behind Judy
x = the number of people in the ENTIRE group (includes Judy)
Then: Troops behind Judy = (x-1)/3
3/4 of ENTIRE group were ahead of those behind Judy ==> x * 3/4
Those that are behind Judy = x * 1/4 = x/4
(x-1)/3 = x/4
4x-4 = 3x (multiply both sides by 12)
x = 4
So, the total number of troops EXCLUDING Judy is 4-1 = 3. There is 1person behind Judy and 2 in front. So looking at the question, 1/3 of the rest of the troop (1 person) is behind Judy, and 3/4 of the entire troop (3/4 * 4 = 3 people, including Judy) are ahead of those behind Judy.
3) Assume: [A] = the rest of the group (excludes Judy)
[B] = ahead of Judy
x = the number of people in the ENTIRE group (includes Judy)
Then: Troops behind Judy = (x-1)/3
Troops in front of Judy = (x-1) * 2/3 (remember: Judy is not counted)
3/4 of {entire group MINUS Judy} were ahead of Judy ==> (x-1) *3/4
(x-1) * 2/3 = (x-1) * 3/4
8x-8 = 9x-9 (multiply both sides by 12)
x = 1
So, the entire troop comprises of only Judy. There is no one behind of her or in front of her. So the total number of troops EXCLUDING Judy is 1-1 = 0. This answer, although correct, is unlikely to be actual answer of the puzzle.
4) Assume: [A] = the rest of the group (excludes Judy)
[B] = ahead of those behind Judy
x = the number of people in the ENTIRE group (includes Judy)
Then: Troops behind Judy = (x-1)/33/4 of {entire group MINUS Judy} were ahead of those behind Judy==> (x-1) * 3/4
Those that are behind Judy = (x-1) * 1/4 = (x-1)/4
(x-1)/3 = (x-1)/4
4x-4 = 3x-3 (multiply both sides by 12)
x = 1
So, the entire troop comprises of only Judy. There is no one behind of her or in front of her. So the total number of troops EXCLUDING Judy is 1-1= 0.
This answer, although correct, is unlikely to be actual answer of the puzzle.
Therefore, the most likely answer for the version 1 of the puzzle is 3, as discussed in Assumption 2 above.
Now let's look at Version 2 of the puzzle. Again, I'll insert additional information in brackets for the parts that are ambiguous. Resting along a path, Judy noticed that she and those behind her made up1/3of the troop (counting Judy?) and that those ahead (of whom?) made up 3/4oftroop (counting Judy?) Counting Judy, what would be the smallest number of hikers that day? The first inserted brackets above shows the first ambiguity. Does the "troop" include Judy or not? We will refer to this ambiguity as [A].
- The second inserted brackets shows the second ambiguity. Is it referring to ahead of Judy" or "ahead of those behind Judy"? This will be ambiguity [B].
- The third inserted brackets shows the same ambiguity as the first one. Due to the way the question is worded, the "troop" here should be the same as the one in [A]. So whatever is assumed for [A] is assumed for this "troop" here. Now, the 4 possible solutions:-
1) Assume: [A] = troop includes Judy
[B] = ahead of Judy
x = the number of people in the ENTIRE group (includes Judy)
Then: 1/3 of the {troop including Judy} ==> x/3
So, Judy and those behind her = x/3
Thus, those ahead of Judy = x * 2/3
3/4 of the {troop including Judy} ==> x * 3/4
So, those ahead of Judy = x * 3/4
x * 2/3 = x * 3/4
8x = 9x (multiply both sides by 12)
x = 0
This answer is impossible as the entire group must consist of at least1person (Judy). So the initial assumption is wrong.
2) Assume: [A] = troop includes Judy
[B] = ahead of those behind Judy
x = the number of people in the ENTIRE group (includes Judy)
Then: 1/3 of the {troop including Judy} ==> x/3
So, Judy and those behind her = x/3
Thus, those behind Judy = x/3 - 1
3/4 of the {troop including Judy} ==> x * 3/4
So, those ahead of those behind Judy = x * 3/4
Thus, those behind Judy = x * 1/4 = x/4
x/3 - 1 = x/4
4x-12 = 3x (multiply both sides by 12)
x = 12
So, the total number of troops INCLUDING Judy is 12. There are 3people behind Judy and 8 in front. So looking at the question, 1/3 of the ENTIRE troop is 4 people including Judy, and 3/4 of the ENTIRE troop (3/4 * 12 = 9 people, including Judy) are ahead of those behind Judy.
3) Assume: [A] = troop NOT includes Judy
[B] = ahead of Judy
x = the number of people in the ENTIRE group (includes Judy)
Then: 1/3 of the {troop excluding Judy} ==> (x-1)/3
So, Judy and those behind her = (x-1)/3
Thus, those behind Judy = (x-1)/3 - 1
3/4 of the {troop excluding Judy} ==> (x-1) * 3/4
So, those ahead of Judy = (x-1) * 3/4
Thus, those behind Judy = (x-1) * 1/4 = (x-1)/4
(Remember: Judy is not included in the troop)
(x-1)/3 - 1 = (x-1)/4
4x-16 = 3x-3 (multiply both sides by 12)
x = 13
So, the total number of troops INCLUDING Judy is 13. There are 3people behind Judy and 9 in front. So looking at the question, 1/3 of the troop MINUS Judy is (13-1)/3 = 4 people (including Judy, since we explicitly added her to get 1/3), and 3/4 of the troop MINUS Judy
((13-1) * 3/4 = 9 people) are ahead of Judy.
4) Assume: [A] = troop NOT includes Judy
[B] = ahead of those behind Judy
x = the number of people in the ENTIRE group (includes Judy)
Then: 1/3 of the {troop excluding Judy} ==> (x-1)/3
So, Judy and those behind her = (x-1)/3
Thus, those behind Judy = (x-1)/3 - 1
3/4 of the {troop excluding Judy} ==> (x-1) * 3/4
So, those ahead of those behind Judy = (x-1) * 3/4
Thus, those behind Judy = (x-1) * 1/4 = (x-1)/4
(x-1)/3 - 1 = (x-1)/4
4x-16 = 3x-3 (multiply both sides by 12)
x = 13
So, the total number of troops INCLUDING Judy is 13. There are 3people behind Judy and 9 in front. So looking at the question, 1/3 of the troop MINUS Judy is (13-1)/3 = 4 people (including Judy, since we explicitly added her to get 1/3), and 3/4 of the troop MINUS Judy
((13-1) * 3/4 = 9 people) are ahead of Judy.
Therefore, the most likely answer for the version 2 of the puzzle iseither12 or 13, based on the assumptions made, as discussed above.
Finally, let's look at Version 3 of the puzzle. Again, I'll insert additional information in brackets for the parts that are ambiguous.
Stopping along the trail to rest, Judy saw that one-third of the rest of the group (i.e.. not counting herself) from the Girl Scouts was behind (whom?) and three-fourths (of what?) were ahead (of whom?). Not counting Judy, what was the total number in her group?
The first inserted brackets above just clarify that "the REST of the group" means Judy is not counted in the number.
The second inserted brackets shows the first ambiguity. Is it referring to "behind Judy" or "behind those ahead of Judy"? This is ambiguity: [A].
- The third inserted brackets shows the second ambiguity. Is it referringto"3/4 of the entire group (including Judy)" or "3/4 of the rest of the group (excluding Judy)"? This is ambiguity: [B].
- The fourth inserted brackets shows the third ambiguity. Is it referring to "ahead of Judy" or "ahead of those behind Judy"? This is ambiguity: [C].
Essentially, this version is the same as Version 1 except that the word "her" is removed to make a new ambiguity. I'm not going to bother doing another full analysis here (and there are 8 of them!), and I'm sure you know how to do them. I haven't analyze this version completely myself, but as this is a reader's own version of the puzzle, I'm going to leave it here for now.
So, in summary, in the original version of the puzzle, 13, 15 or 9 is not a correct answer, but 4 is. In the second version, both 12 and 13 are correct, but note that the 2 versions are very different and yield completely different answers. So one cannot substitute the first version with the second.
So, what does all these come down to? The "problem" of this puzzle is that it is not well-defined. I agree that a puzzle can allow for playing on words, but make sure it can only be played one way, and not multiple ways. Otherwise, there will be a lot of possible answers, and each of them is as accurate as the other, like this puzzle here.
Phew!
Phew indeed!! And with this explanation, "the books are closed on this one!!"
