Quite possibly the most intimidating problem on the GRE contains strange symbols: @, #, *, or a black circle often accompany these problems. Many recoil in horror thinking – I’ve never learned that before! (Or perhaps more aptly, what the @#?!)
But don’t despair – the symbols are completely arbitrary and are defined on the spot by the GRE. Here is an example:
#x# = 
. What is the value of #3# – #2#?
Again, the pound sign surrounding the number has no mathematical meaning outside the problem. For the question, you simply want to follow the rules. Here, wherever we see a number between the pound sign, such as #3#, we want to refer back to #x# = . The 3 essentially is taking the place of the x. So if #x# = , then #3# = 
.
Now do the same for #2#: 
.
So #3# – #2# = 
Now let’s try another one. This time, though, I am going to put a little spin on it.
n^^ = 
, where n is a positive integer. For how many values of n is n^^ less than zero?
- 1^^
- 1^^ – 2^^
- 3^^
- 3^^ + 4^^
- 5^^ – 2^^
First off, note that ! is the factorial sign. It is not a strange symbol, but standard mathematical notation.You should quickly see that after n = 3, n^^ is going to yield a positive result. For example, 
, so 4^^ 
. So when n is greater than or equal to 4, n^^ is greater than zero.
Be careful: 1^^ = 0, so it is not less than zero. Therefore, there are two values (2, 3) for which n^^ is less than zero.
When we look at the answers, 2 is not among them. Instead, the strange symbol ^^ has been reintroduced. Therefore you have to figure out which answer choice equals 2, the number of values of n that are less than zero.
The answer is (B), which gives us 1^^ = 0 minus 2^^ = -2, so 
Okay, that was a tough one. Let’s make the problem easier, while adding a layer of complexity – the embedded strange symbol.
If x is even, @x = 
; if x is odd, @x = 
. What is the value of @(@(@5)?
- 21
- 40
- 63
- 117
- 140
Notice I’ve used the strange symbol three times. Don’t worry – just follow the operation (the technical name of this process). Taking the problem apart one step at a time, we get @5 = 8. @8 = 21, and @21 = 40. Just like that, B.
Takeaway
Don’t be freaked out by strange symbols on the GRE. The question will always clearly define the symbol for you. Carefully follow the steps to the correct answer.
I am not satisfied with the following question and explanation….. can you please elaborate on it ?
If x is even, @x = ; if x is odd, @x = . What is the value of @(@(@5)?
21
40
63
117
140
Notice I’ve used the strange symbol three times. Don’t worry – just follow the operation (the technical name of this process). Taking the problem apart one step at a time, we get @5 = 8. @8 = 21, and @21 = 40. Just like that, B.
Sure!
@5 = 2(5) – 2, where 5 takes the place of x in the operation @x = 2x – 2.
So @5 = 8.
8 is an even number so we use the other operation: @8 = 3(8) – 3 = 21.
21, an odd number, must use x@ = 2x – 2. So @21 = 2(21) – 2 = 40 (B).
Hope that helps!
Hi Chris,
n^^ =n!-n^2 , where n is a positive integer. For how many values of n is n^^ less than zero?
Lets consider ans choices B and C,
B. 1^^ = 0 minus 2^^ = -2, so 0-(-2)=2, therefore the result is positive.
C. 3! – 3^2 = 6 – 9 = -3, Therefore the result must be “C”.
Why do you say that its “B”?
Hi Nevin,
This question is a little tricky. Let’s take apart one step at a time.
First, the question is asking for the number of values of n that are negative. Even though the values of n are negative, the number of values (i.e. 1, 2, 3, etc) is a positive number. Thus, there are two values of n (2,3) in which n^^ is negative.
Secondly, the answer choices are expressed as ^^. So we have to find which one of the answer choices yield the number 2. Only (B) does so. As you pointed out 1^^ – 2^^ = 2.
Hope that helps!
Got it, Thank you… I was asinine
Hi Chris,
Would you please do this problem for me?
Q: A pair of dice are so produced that when rolled simultaneously, exactly one of them always shows a prime number, what is the probability that the products of the numbers on the dice is even?
(a)7/9
(b)1/2
(c)2/9
(d)7/18
(e)7/36
When two dice are rolled, there are 6*6=36 possible outcomes. Now, the prime numbers among numbers 1,2,3,4,5,6 are 2,3 and 5. So on exactly one dice, a prime number turns up.
If 2 turns up on one, the other dice can show 1,4 and 6. (Only one number is prime).
If 3 turns up on one, the other can show 4 and 6 (remember the product should be even).
If 5 turns up on one, the other can show 4 and 6.
Hence we have the pairs – (2,1), (2,4), (2,6), (3,4), (3,6), (5,4), (5,6).
That’s totally 7 pairs. And the same thing for the other dice as well. Hence that’s a total of 7+7= 14 favorable outcomes and 36 possible outcomes. 14/36 = 7/18
Answer = (d)
Ajey,
Great response!
Thanks for the clear step by step approach.
Thanks Ajey, GRE problems always seem so easy …..After we get to know how its done
Nevin,
Check out Ajey’s great response!.
One thing I would elaborate on is the last step. The reason the outcomes could apply to the other die, is we never stipulated which of the two dice landed on a prime.
Because either of the two could be the prime (but just one at the same time), we multiply by 2.
Hi Chris,
I understand the other problems, but am struggling big time with n^^ = , where n is a positive integer. For how many values of n is n^^ less than zero? First, how did you determine from the questions that “You should quickly see that after n = 4,” there are no numbers involved. Did you just guess at a number or are you looking at possible answers and plugging those numbers in? I will be honest, I am totally lost on this question and would love further clarification.
I have another problem that I was given on my practice test booklet that I was struggling with as well which led me out to seek some answers on the internet.
Hi Patty,
The part that read, “you should quickly see…” should read including n = 4, not after n = 4. I’ve made the necessary changes. Let me know if that helps. Also, read the response above Nevin’s comment above.
If that doesn’t help, I’d be happy to clear things up.
As for the practice problem, I’d love to answer it. Out of curiosity, does it include strange symbols? That way, it would be relevant to the post.
Either way is fine though.