# Mensa IQ test question - lines in squares

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

• When I took my Mensa test, we weren't allowed to copy any of the questions. Where did you get this from? ;-) Aug 10 '15 at 14:49
• @DevSolar- Photgraphic memory? Aug 10 '15 at 17:16
• 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. Aug 10 '15 at 20:34
• 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... Aug 11 '15 at 9:08
• None of them are missing; they're all right there. Aug 16 '15 at 4:46

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: • How does Row 2 work? Aug 10 '15 at 14:00
• Really nice find, Bailey. Aug 10 '15 at 14:01
• 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. Aug 10 '15 at 14:21
• 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 Aug 10 '15 at 17:33
• well done. bravo. Aug 11 '15 at 13:27

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

• 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". Aug 10 '15 at 22:47
• 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. Aug 11 '15 at 0:39
• @FMaz008 - It's very smart; it's just not what Mensa want.
– h34
Aug 11 '15 at 6:50
• @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!" Aug 11 '15 at 7:11
• @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". Aug 11 '15 at 22:22

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.

• If this was Tic Tac Toe, the game would have been a draw by at least the previous turn.
– Mast
Aug 12 '15 at 22:26
• @Mast Obviously, but what kind of person doesn't finish filling it out anyways? Aug 12 '15 at 22:41

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:

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.

• 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". Aug 11 '15 at 15:43
• 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 7:58
• 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 8:03
• 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...) Aug 12 '15 at 8:30
• (...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. ;-) ) Aug 12 '15 at 8:33

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.

• +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 7:48
• The placement within columns is irrelevant to this answer; this indicates that you have ignored information. Aug 12 '15 at 14:43
• @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. Nov 23 '16 at 18:27