Data Sufficiency

A list contains twenty integers, not necessarily distinct. Does the list contain at least two consecutive integers?

(1) If any single value in the list is increased by 1, the number of different values in the list does not change.

(2) At least one value occurs more than once in the list.

I have to confess, this one was tough for me. The most straightforward way to test the statements is to use very simple sets.

S1 If any single value in the list is increased by 1, the number of different values in the list does not change.

Let's try a set of consecutive evens: 2, 4, 6, ... We'll have 20 different integers to start. If we increase any of them by 1, we'll still have 20 different integers, so we can use this set. And here we do not have any consecutive integers, so we have a NO.

Let's try to get a YES; this was a challenge.
Imagine a set like: 1, 1, 2, 4, 4, 5, 5, 6 (for simplicity's sake, we'll make a set with 8 integers, but the logic will work for a set with 20 integers too.)
We're starting with 5 different integers. If we increase a 1 to to a 2, we'll have 1, 2, 2, 4, 4, 5, 5, 6 --> still have 5 different integers. If we increase the 2 to a 3, we'll have 1, 1, 3, 4, 4, 5, 5, 6, we still have 5 different integers. If we increase the 4 to a 5, we'll still have 5 different integers. Same if we increase the 5 to a 6 or the 6 to a 7. So this set gives us a YES, we do have some consecutive integers here. (Getting a set to work was a little brain-melting.) S1 = Not Sufficient.

S2: This is easier:
First set: 2, 2, 4, 6, 8 .... No consecutive = NO
Second set: 2, 2, 3, 4, 5.... Consecutive integers = YES
So S2 is not sufficient.

Together: Let's go back to the sets I used for S1: Set One: 2, 4, 6, 8... And Set Two: 1, 1, 2, 4, 4, 5, 5, 6. Only the second set will work for our together scenario because the first one has no repeated elements. So I know I can get a YES using Set Two. The question is, can I get a NO, if both statements are true at the same time?

And I don't see how: A set like this: 2, 2, 4, 4, 6, 6 would violate the first statement: changing the 2 to a 3 would increase the number of different integers in the set. Not usable. The only way to get a set that satisfies both is to have some consecutive elements, so together the answer must be YES. Answer = C

(Confession: this took me longer than 2 minutes. Brutal question.)

j_shreyans wrote:A list contains twenty integers, not necessarily distinct. Does the list contain at least two consecutive integers?

(1) If any single value in the list is increased by 1, the number of different values in the list does not change.

(2) At least one value occurs more than once in the list.

Whoa got smoked.

Another easy case that satisfies statement 1 is as follows:
{1, 1, 1...1, 1, 2}
Here, the list contains two different values (1 and 2).
If 1 increases to 2, or if 2 increases to 3, there will still be two different values in the list.
In this case, the list contains two consecutive integers -- 1 and 2 -- so the answer to the question stem is YES.

Another easy case that satisfies statement 1 is as follows:
{1, 1, 1...1, 1, 2}

Now that is beautifully simple. Nicely done, Mitch.

GMATGuruNY wrote:Another easy case that satisfies statement 1 is as follows:
{1, 1, 1...1, 1, 2}
Here, the list contains two different values (1 and 2).
If 1 increases to 2, or if 2 increases to 3, there will still be two different values in the list.
In this case, the list contains two consecutive integers -- 1 and 2 -- so the answer to the question stem is YES.

Hi Mitch,

About statement 2, since at least 1 value occurs more than once and there is no 'at most' condition-

[1,1,1,..1] or [1,1,2,2..]

Statement 2 would be insufficient..?

Thanks,
Mallika

mallika hunsur wrote:

GMATGuruNY wrote:Another easy case that satisfies statement 1 is as follows:
{1, 1, 1...1, 1, 2}
Here, the list contains two different values (1 and 2).
If 1 increases to 2, or if 2 increases to 3, there will still be two different values in the list.
In this case, the list contains two consecutive integers -- 1 and 2 -- so the answer to the question stem is YES.

Hi Mitch,

About statement 2, since at least 1 value occurs more than once and there is no 'at most' condition-

[1,1,1,..1] or [1,1,2,2..]

Statement 2 would be insufficient..?

Thanks,
Mallika

Your reasoning is sound.

Statement 2: At least one value occurs more than once in the list
Case 1: {1, 1, 1...2}
In this case, the list contains at least two consecutive integers, so the answer to the question stem is YES.
Case 2: [1, 1, 1...1}
In this case, the list does not contain at least two consecutive integers, so the answer to the questions stem is NO.
INSUFFICIENT.