# Mensa Norway last 3 questions

I had no problems with solving first 32 questions from http://test.mensa.no but the last 3 questions are totally bugging me out even knowing the answers. Could anyone help me getting the logic behind them?

E A D

For the last one:

All the diagonals and broken diagonals have a shape in common. For those going NE we have:
And for those going SE we have:

For the second one:

The black balls follow the rule of diagonals going NE described above. The white balls follow a separate rule:

There are two white balls in every cell and they move in a counter-clockwise direction in the column they are in, as shown by the yellow balls above. If a white ball is at the same location as a black ball, only the black ball is shown.

The link provided by @Duck in the comments gives the answer to the first problem, so all three problems should now be answered.

• I saw this answer in comments but tbh I don't get the explanation for the first riddle. Could you explain it better for me? – Michał Pilarek Sep 19 '19 at 8:32
• Look at the $3$ NE diagonals and assume the triangles can flip across the line they are connected to. Only one flip is allowed per row change and you start in the top row, moving down. Assume also that a striped triangle flipped onto a white triangle gives a striped triangle. Hope that helps! – Jens Sep 20 '19 at 1:55
• Ok, now I get it – Michał Pilarek Sep 20 '19 at 11:26
• The answer could also be E for the last one since because each row has in total then (of the rectangle) two tops, one bottom and one left and one right. This is why iq tests often annoy me because you can have a logically consistent answer and still be wrong. Oh well they are for mainlynfor people who don't know that they are smart (which probably means they are not if they take it). – Deepthinker101 Sep 20 '19 at 23:13

3 is:

D. The matrix can be interpreted as a determinant, with the 'factor' of a cell being formed from $$(TL \land BR) \lor (TR \land BL)$$, where $$\land$$ returns the common components of the two images, and $$\lor$$ returns all components of the images.

The correct answer for the first one is

E. Look at the 2nd block and see the figure which is shaded out and now see the other two figures, you will see that the other two figures are just opposite/reverse of each other. Now come to the 3rd block and see the figure which is shaded out and see the figure which is not shaded out. So the correct answer should be the figure which is the opposite/reverse of the non-shaded figure in the 3rd block, which is option E.

For the first problem a.k.a problem #33 I looked it up in downward direction. For each column we can see that there is a triangle that never “moves” (red circle) and that's our reference point and it follows the clockwise pattern . So we can expect that in the third column there will be a plain triangle at the upper left corner.

The rest is finding the alternating triangle pattern start from the first column and I guess you can just follow it with few adjustments for the rest of the columns.

I got A and here is my reasoning (of course, no guarantee it's correct or even CLOSE to the most logical solution):

Before we delve into the main explanation, let us establish these two main variables:

1. The number of triangles ABOVE and BELOW the line
2. The number of triangles BLACK and WHITE

For the following equations, we subtract the LARGER figure by the SMALLER one.

Focusing on the 1st ROW, we can see there are 5 triangles on the bottom half and 4 on the top. 5 - 4 = 1 2nd ROW: 4 - 3 = 1 3rd ROW (Replace the question mark with A) 4 - 2 = 2

Now let's trying adding the answers of these three equations together: 1 + 1 +2 = 4

Keep the answer 4 in mind as we check the same variable for the columns

Observing the 1st COLUMN, there are 4 on the BOTTOM and 2 at the TOP, so I would answer the equation 4 - 2 = 2; The LARGER figure goes FIRST in these subtraction equations. 2nd COLUMN: 5-3 = 2 3rd COLUMN (once again replacing the missing tile with the one from A): 4 - 4 = 0

Time to add these 3 answers up again! 2 + 2 + 0 = 4

As you can see, the sum for these TWO groups of THREE subtraction equations are EQUAL!!!!

Also notice how there is a PAIR of equal answers to the subtraction equations in both the ROWS and COLUMNS (1 for the ROWS, 2 for the COLUMNS). I'll attempt to postulate why I don't think this is a coincidence later, but for now, let's move on the 2nd VARIABLE!

Let's recap the second variable: "The no. of triangles that are BLACK and WHITE"

Why don't we take a shot at doing the same style of subtraction equations for the ROWS and COLUMNS once again, except this time for the no. of BLACK and WHITE triangles?

Taking a look at the 1st ROW, I can see 6 BLACK and 3 WHITE. May someone double check for me? 🤣 Let's perform subtraction yet again, remembering to put the larger figure first as we'd done for all our earlier equations. 6 - 3 = 3

2nd ROW: 5 - 2 = 3

3rd ROW (remember once again to replace tile A into the unknown space): 3 - 1 = 2

Adding the answers to these 3 equations gives us 3 + 3 + 2 = 8 (EIGHT)! Keep this number in mind as we move onto the COLUMNS

1st COLUMN: 4 - 2 = 2 2nd COLUMN: 5 - 3 = 2 3rd COLUMN (Testing tile A): 6-2 = 4

Let's sum it up! 2 + 2 + 4 = 8. THE ANSWER IS EIGHT AGAIN!!!!

Also notice how there's a pair of common answers in BOTH the ROWS and COLUMNS, just as with our 1st variable?! TAKE A LOOK! For the ROWS, there are a PAIR of 3s and in the COLUMNS, there are a PAIR of 2s! Just in case you forgot, here's the first variable: "The number of triangles ABOVE and BELOW the line".

It appears to me adding the answers of the subtraction equations from The ROWs and the COLUMNs separately gives us the same ANSWER!!!

Remember how I postulated at the beginning that I don't think the pairs of equal answers for each triplet of subtraction equations are a coincidence?

My reasoning is a PAIR is composed of TWO entities. There are ALSO TWO pairs of equal numbers for each variable. Summing up the answers to the subtraction equations in the ROWs and COLUMNS separately for the 2nd VARIABLE gave us 8 for BOTH, which is DOUBLE 4 (the answer we obtained summing up the answers of the subtraction equations in either the ROWS or COLUMNS individually for the 1st VARIABLE!

My final answer is A because none of the other tiles allow these rules to apply for 100% of cases.

Hope my response helped! 🤗👍🏼

If you (reading this right now) manage to poke holes through this theory, feel free to adduce a more logical explanation!