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Which of the 8 tiles below is missing?

I've been trying to figure out the answer to this question for quite some time. Can anyone spot the pattern?

Source: https://www.bergmandata.com

bobble
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Mathxx
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    When I took my Mensa test, we weren't allowed to copy any of the questions. Where did you get this from? ;-) – DevSolar Aug 10 '15 at 14:49
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    @DevSolar- Photgraphic memory? – chasly - supports Monica Aug 10 '15 at 17:16
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    It wasn't clear to me until reading @BaileyM's answer that the numbered tiles were the multiple-choice answers. I think it would be good to clarify that, so that people like me don't think, incorrectly, that the numbered tiles are also part of the puzzle statement. – Jashaszun Aug 10 '15 at 20:34
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    Weird that this has been #1 on HNQs for so long. It's a very nice question and answer, but with only one answer and few edits, I can't understand how it became so 'hot'! – Rand al'Thor Aug 10 '15 at 21:11
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    @Jashaszun Guessing that fact might be part of the IQ test ;-) – Paŭlo Ebermann Aug 10 '15 at 21:33
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    The real challenge would be to create an answer that shows that you could think of 8 different patterns each yielding a different tile as the solution... – Jeroen Aug 11 '15 at 09:08
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    @Jeroen - I think that would be an excellent challenge. We have already got answers for tiles 2, 3 and 5. Hopefully people will post answers for the other five tiles. – h34 Aug 11 '15 at 12:21
  • @Jeroen yup. Any other way it'd be yet another "guess what the author had in mind" puzzle. –  Aug 13 '15 at 19:47
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    @DevSolar Your way of communicating is exactly why a lot of people don't want to join Mensa: you prefix EVERYTHING you say with something to make it known that you are a Mensa member. – user2520938 Aug 15 '15 at 17:34
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    @user2520938: Which would be funny, because I am not a Mensa member. I took the test, and passed, but that doesn't mean I joined, or feel any association with that organisation. But they are the ones organizing these tests, and the OP stated this is a Mensa IQ test question, so.... trying to find fault where there isn't, perhaps? – DevSolar Aug 16 '15 at 00:23
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    None of them are missing; they're all right there. – Dax Fohl Aug 16 '15 at 04:46

7 Answers7

208

I think the answer is

Tile 5

Because it looks to me like

Reading from left to right, the lines continue where the previous square left off, to make two long connecting 'strings' of sorts. The eighth square leaves off in the top righthand corner and about 80% of the way down the bottom side, which is where tile 5 would connect to.

Here is an image of the final solution:

Mensa Squares

Bailey M
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    How does Row 2 work? – Raystafarian Aug 10 '15 at 14:00
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    Really nice find, Bailey. – VictorHenry Aug 10 '15 at 14:01
  • I think Row 2 works vertically downwards from Row 1's 3rd panel and continues until Row 3's first panel. – Doge Aug 10 '15 at 14:01
  • @Raystafarian, row 2 still just connects up to the previous square. It reads like a book, left-to-right, then down a line, etc etc. So the leftmost square of Row 2 connects to the bottom of the rightmost square of Row 1, since that's where the lines meet up. – Bailey M Aug 10 '15 at 14:03
  • @Raystafarian Basically whenever a line touches the right wall. a line will appear on the same height on the left wall of the next square. Whenever a line touches the bottom wall it will appear on the same width on the top wall on the next square. – Ivo Aug 10 '15 at 14:16
  • Ah, not at all what I thought you were describing +1 – Raystafarian Aug 10 '15 at 14:18
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    LOL I was like Tile 5 cause of basically the reason you gave, I greatly enjoy how you enhanced the answer into the full picture, the extra effort just makes the question and answer so much more complete and beautiful. – Going hamateur Aug 10 '15 at 14:21
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    I missed the connector from 1,3 to 2,1 and 2,3 to 3,1 - though the solution can be found if you take them as 3 separate series, as I did – Jason Aug 10 '15 at 17:33
  • If we allow lines to stop at corners, as if they go off the strip, we can position all 3 rows horizontally, like this. – h34 Aug 11 '15 at 13:18
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    well done. bravo. – Jason P Sallinger Aug 11 '15 at 13:27
  • This is definitely the answer that I saw as I was reading the tiles. The pattern was obvious before I even got to the fourth tile. I especially liked your succinct diagram. – Ian MacDonald Aug 12 '15 at 14:44
  • damn, I saw this pattern for the first row but my brain never connected the first and second row properly. – DanielST Aug 12 '15 at 18:29
  • Generally the point of these questions is to realize that all answers different to nb 5 (in this case) is wrong. – balazs.com Oct 26 '19 at 23:02
  • @Bailey M, May I ask what software you used to create the final image ? – Hemant Agarwal Jun 17 '20 at 21:37
26

I think the answer is

The one given by @Bailey M, but I would like to offer another answer...

Because I figured that

It would be fun to have an alternative solution :-). With this solution, the supposed rules for tiles are that:

1. the third column tiles' lines always "indicate" adjoined sides;
2. the third column tiles' lines always touch exactly one corner;

Meaning that

Tiles 1, 2, 7, and 8 are out because of rule 1;
Tiles 4, 5, 6, (and 7) are out because of rule 2;

And thus that

it must be Tile 3!

Here's an image to visualize that

visualization of the solution

Community
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Jeroen
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    Which is why I hate this sort of problem--there are usually valid arguments for multiple answers and it's pretty hard to define "best". – Loren Pechtel Aug 10 '15 at 22:47
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    Loren Pechtel: I often run into theses problems where I can explain a simple logic for the answer I select in IQ tests, but it seem 5% of the time I still get it wrong. Duh. Thinking out of the box is not smart. – FMaz008 Aug 11 '15 at 00:39
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    @FMaz008 - It's very smart; it's just not what Mensa want. – h34 Aug 11 '15 at 06:50
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    @h34 Exactly. I can't see a puzzle like this without thinking of the story in one of Douglas Hofstadter's books about when he gave a lecture about training computers to solve "what number comes next in this sequence?" problems, and Richard Feynman was sitting in the front row. Every time he put up an example sequence on the board and asked the audience, "What number comes next?" Feynman shouted "Nine!" – dodgethesteamroller Aug 11 '15 at 07:11
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    Same comment here as for h34's answer: This completely ignores the information from column 1 and 2. In the context of other "identify the tile" questions, this is unlikely. – DevSolar Aug 11 '15 at 15:51
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    @dodgethesteamroller You wouldn't mind explaining that one? Don't I feel stupid :-) – Voo Aug 11 '15 at 17:49
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    @Voo I would guess the point is that you can always come up with a "pattern" (of arbitrary complexity) that will give any answer you like as the next in the sequence. An extreme example of this is overtraining with artificial neural networks - the ANN mistakes the training data itself as being the pattern, instead of identifying the more generalised pattern. For example, if I ask you what number is next in the sequence "1 2 3", you might say "4", but Richard Feynman might say that for every 4th item you add 6 instead of 1, justifying his "9". – JBentley Aug 11 '15 at 22:22
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    That solution does not make to use columns 1 and 2 at all, or am I missing it? I think one of the point of Mensa tests is that they do not give you irrelevant information, everything must fit. – Andrew Savinykh Aug 12 '15 at 01:25
  • I'm not sure I follow the rule #1. How do the two tiles in the third column relate to each other via this rule? What does "indication" mean in this context? – Ian MacDonald Aug 12 '15 at 14:37
  • @IanMacDonald The two lines inside the tile cross the borders of that tile at the red circles. Those two lines thus "indicate" or "point at" a certain side of the tile (marked in blue in the picture). The two borders that are "indicated" can be either opposite, or orthogonal (as they are in the third column). – Jeroen Aug 12 '15 at 15:52
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    @JBentley Well put. Without context, there's no reason to prefer one sequence of numbers over another (since any "next" number can be justified by a new arbitrary rule). The number-sequence game is meaningless; it's a seductive but ultimately invalid analogy to what scientists do when they look for recurring patterns of phenomena in nature and then try to infer universal laws that explain those observations. So Feynman was making a philosophical point. Coming back to the OP's question, there's in general no philosophically valid reason to prefer one answer to the Mensa puzzle to another. – dodgethesteamroller Aug 12 '15 at 19:23
21

You're all thinking to hard. This is just a game of Tic Tac Toe. Since it is the turn of X, tiles 1, 4, 5, 6, and 7 would all be correct. I personally would go with 7 because it's a well drawn X.

Lampost42
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13

I think Bailey M has got the intended answer.

But another answer that works is

Tile 2

Reasoning:

Notation
In each square, label the corners A-D, running clockwise from the top left. By "line" we mean a line inside a square. Call a line "par" if it is parallel to a side of the square. Call a corner "visited" if it is touched by a line.

First fact
Consider the distribution of par lines by row. In rows 1-2 the totals are the same: each row contains exactly 1. To continue the pattern of sameness, row 3 must also contain exactly 1. Of all the possible missing tiles, the only one that contains exactly 1 par line is tile 2.

Second and corroborating fact.
Now consider the distribution of par lines by column. In columns 1-2 the numbers are different: there are 0 in column 1 and 2 in column 2. Neither of the known tiles in column 3 contains a par line, so the total in column 3 can only be 0 or 1. To continue the pattern of difference, the number must be 1, so again the answer must be tile 2.

Third and supporting fact
Each fully known column contains exactly 2 squares in which the same corner is visited: in column 1, that is corner B; in column 2, corner C. So we need a tile that continues that pattern in column 3. The visited corners in the known tiles in column 3 are A and C. This restricts the possibilities for the missing tile to tiles 2 and 5. So when we choose tile 2, this pattern also continues.

Diagram of this solution:

diagram

Of course this answer doesn't use all of the given information. For example, the horizontal line in the top middle tile could be translated upwards and everything would still apply. But the same is true of what appears to be the intended solution. In that solution, it doesn't matter whether two lines in a square cross or don't cross, so long as they start and end so as to connect with the lines in adjacent squares. One could also say that going downwards for row 2 but left-to-right for rows 1 and 3 is a bit arbitrary. Our basis for deciding what we think is the best solution is subjective, or at least would take a lot of definition to be made objective.

h34
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    Nice reasoning. But when looking at this and Bailey's solution, his is using more information from the tiles. And although my test has been over ten years since, I think I remember that this group of question always used all available information from the tiles; there was no "noise". – DevSolar Aug 11 '15 at 15:43
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    I accept that Bailey M's solution uses more of the given information (for a handwaving definition of "more"), is neat, and is almost certainly what Mensa wanted. But the puzzle doesn't ask for the solution that uses the most information - it just says find "the" tile that fits, suggesting that we stop when we've found one. Kolmogorov and Solomonoff aside, I don't think there's a rigorous definition of "amount of information" that's relevant here. @ANil GAdiyar's solution below doesn't use where every line ends, but uses whether they cross; the opposite is true of Bailey M's solution. – h34 Aug 12 '15 at 07:58
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    No solution so far has used all the information, even in an unpedantic and down-to-earth sense. One thing I find very interesting here is the role of people's knowledge of what Mensa want, which says a lot about IQ tests. (Correction to my previous comment: Bailey M's solution uses where all lines end except on the left of tile (1,1) and the bottom of the solution tile.) – h34 Aug 12 '15 at 08:03
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    The way a Mensa test works (at least, how it worked when and where I took it) is this: You get a booklet with the questions, subdivided in sections. One section about puzzles like these, one about memorizing, one about words... you get the idea. Each section starts with examples, which are explained by the person giving the test, so you know what kind of solutions are expected. Only when everybody in the room is ready, the page is turned to the real questions, and the clock starts ticking. When the time is up, the next section is being explained. (ctd...) – DevSolar Aug 12 '15 at 08:30
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    (...ctd) So you get a pretty good idea of what kind of answer the questions are looking for. Besides, you cannot realistically solve all the questions correctly in the time frame, so skipping a dubious one doesn't really hurt. If this one question is the only one you got wrong, you scored off-scale high and should take the next harder test anyway. Tests are only precise for ~ 15 IQ points around a given mark, and rapidly lose significance outside that range. (Which makes people bragging about the 200+ IQ value they got from some website test targeting the 85-115 IQ range so funny. ;-) ) – DevSolar Aug 12 '15 at 08:33
  • @DevSolar Interesting and funny :D – ABcDexter Jun 18 '16 at 18:41
5

answer is

2

because

row 1 has 3 kinds of design:

a) one with straight line;
b) one with two lines intersecting;
c) and one with a line starting from centre of a tile.

enter image description here

Mr Pie
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ANil GAdiyar
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    +1. A very nice solution! The amount of "information" in favour of tile 2 is increasing :-) But you should change "straight line" to "line parallel to a side of the square" and "from centre" to "from near the middle of a side". – h34 Aug 12 '15 at 07:48
  • The placement within columns is irrelevant to this answer; this indicates that you have ignored information. – Ian MacDonald Aug 12 '15 at 14:43
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    @IanMacDonald - All answers so far have ignored some information. For example, the top answer ignores whether the lines in a square cross or don't cross. – h34 Aug 12 '15 at 22:27
  • Here's my reason for choosing 2 (which I admit doesn't take into account all the information). In each row there is exactly one square consisting of a)1 rectangle, b)1 triangle, and c)1 trapezoid. Squares 7 and 8 don't satisfy this condition. Therefore, unknown square 9 must contain 1 rectangle, 1 triangle, 1 trapezoid. Choice 2 is the only possible answer consisting of 1 rectangle, 1 triangle and 1 trapezoid. Therefore it the only possible valid answer for 9 which obeys this rule. – idiotprogrammer Nov 23 '16 at 18:27
5

The answer is...

Tile 5

Because...

Tile 5 is the only one represented on the bottom that doesn't have a similar tile on the top. I'm not sure why there's so much reading into this... All the other tiles have a similar one on the top even if you have to flip it, etc.

Pilot
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3

I think the answer should be

(3) because for every figure you see in a row one end of two lines lies on one side and other end on another side and also 2 out of 3 of such figures have such sides opposite to each other and one has it adjacent (note having line start at edge should be interpreted as in a favor of this hypothesis i.e. benefit of doubt goes to me). Same pattern is seen in row 2 and hence extrapolating it, I eliminate the eight-options to arrive at (3) which satisfies this rule.

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