The total of the row sums must be equal to the total of the column sums (since each square gets counted once), so the sum of the numbers in the first column is simply $40 + 32 + 31 + 29 - 32 - 31 - 35 = 34$ .

BY noticing the pattern we find out:

In the first row, There are 4 circles, And their sum is 40. Let Circle Be C, Triangle be T, Pentagon be P and Heart be H.

4C=40.

Therefore C =10.

$\color{#3D99F6}{\text{Circle}}$ = 10.

In the fourth column, There are 2 circles and 2 triangles, and their sum is 32.

2C+2T=32.

As we know, C =10.

Therefore:

20+2T=32

2T=12

Therefore, T=6.

$\color{#D61F06}{\text{Triangle}}$ = 6.

In the third row, There is 1 Circle, 1 Triangle, 1 Heart and 1 Pentagon.

Their SUM is 31.

C+T+H+P=31.

10+6+H+P=31

H+P=31-16

H+P=15.

In the fourth row, There is 1 Triangle, 1 Pentagon and 2 Hearts.

Their SUM is 29.

T+P+H+H=29.

6+H+P+H=29

As we know H+P=15.

Therefore: 6+15+H=29

H=29-21

H=8.

$\color{#E81990}{\text{Heart}}$ = 8.

H+P=15.

8+P=15

P=15-8

P=7

$\color{#20A900}{\text{Pentagon}}$ = 7.

Therefore the missing number = $\color{#3D99F6}{\text{Circle}} + \color{#3D99F6}{\text{Circle}} + \color{#D61F06}{\text{Triangle}} + \color{#E81990}{\text{Heart}}$ =(10+10+6+8)

= $\boxed{34}$ .

Answer Guide:

4 Circles = 40

Therefore Circle = 10

but 2 Circles and 2 Triangles = 32

Therefore 2 Triangles = 12

Therefore Triangle = 6

Circle+Circle+Pentagon+Heart= 35

But 2 Circles=20

So Pentagon+Heart = 15

Triangle + Pentagon +2 Hearts= 29

Triangle = 6

Pentagon +2 Hearts = 23

Therefore, from above, Heart = 23-15 = 8

8+Pentagon = 15

Pentagon = 7

2 Circles + Triangle + Heart = ?

10+10+6+8 = 34

?=34

Therefore 34 is the answer.

Each circle is 10 since the first row says 4 circles = 40.

Each triangle is 6 since the second row says 2 circles + 2 triangles = 32; so [32 - (10 + 10)] / 2 = 6.

If pentagon is y and heart is x, then the value of y is 15 - x, since 31 - 6 - 10 = 15.

The last row can be interpreted as an equation 2x + y + 6 = 29.

Thus,

2x + (15 - x) + 6 = 29

2x - x = 29 - 15 - 6

x = 8; y = 7

So, first column would be 10 + 10 + 6 + 8 = 34

Same as Arpit.

Circle = 40/4 = 10; Triangle = (32 -(2 x circle))/2 = 6; pentagon + heart = 31 - (circle+triangle) = 15; 2 x heart + pentagon = 29 - triangle = 23; heart = 23 - 15 = 8; pentagon = 15 - heart = 7; 2 x circle +triangle + heart = 20 + 6 + 8 = 34

The first row has all blues meaning equal value.The value of each 40/4=10. With value of blue known, value of red is 8. in third row, the combined value of blue n pink is balancing figure of 15.In last row, the value of pink after assigning value of blue and combined value of 15, the value of pink is known as 6. With value of blue, red and pink known, he total value of first column is hence 10,10,8 and 6 together or 34.

when you look at the first row, you instantly think the circle = 10 when the answer is 40. then when you look at the second row, knowing that the circle =10, you can determine that the triangle = 6 when you have 2 circles and 2 triangles add up to equal 32 (10+10+6+6=32). now when you look at the second column from the left to right, you subtract 20 from 35 which will get you 15. after finding out that the pentagon and the heart should equal 15, you can use the bottom row to find out what heart is just by adding up the sums of 15. doing this, you will find out the pentagon= 7, and the heart=8. now that you know all the numbers of each shape, you can easily solve the first column (from left to right), and the get the answer 34

Reduce the shapes to numbers to simplify. What divided by 4 is 40? 10.

So Circle equals 10.

There are 2 circles and 2 triangles in row 2. Circle is 10, leaving 12 leftover after subtracting 2x10. 12 divided by 2 remaining slots is 6.

So Triangle equals 6.

That's where the simple part stops. (At least for me)

I made a chart on a piece of paper, filling in circles as 10 and triangles as 6. Then subtracted the value of each triangle that appears in each row and column, and subtracted the value of each circle that appears in each row and column, from the numbers that indicate the total value.

That leaves 15 left to determine in row 3 and columns 2 and 3 , and 23 for row 4, simplifying the question down to:

What groups of 2 numbers added together make 15, and which of those groups, when one of the 2 numbers is added again, equals 23? The number that is added twice is the value of Heart, which is necessary to determine the total of column 1. We can rule out 0 as either value, since 15 + 15 + 0 is greater than 23 and 15 +0 + 0 is less than 23.

This leaves the possible numbers for Heart + Pentagram as:

1 + 14, 2 + 13, 3 + 12, 4 + 11, 5 + 10, 6 + 9, 7 + 8

Of these possibilities, only 7 + 8 + 8 where 8 is Heart and 7 is Pentagram allow the boxes to be filled in without changing the total value of any of the existing rows.

I'm terrible at explaining things to people...if this doesn't make sense I've likely left out a step I took for granted.

Circle = C Triangle = T Pentagon = P Heart = H

Top row: 4C = 40 Therefore C = 10

Second row: 2C + 2T = 32 2(10) + 2T=32 20 + 2T = 32 2T = 12 Therefore T = 6

Third row: T + P + H + C = 31 6 + P + H + 10 = 31 16 + P + H = 31 P + H = 15

Bottow row: 2H + P + T = 29 2H + P + 6 = 29 2H + P = 23

P + H = 15 2H + P = 23

P + H + 8 = 23 2H + P = 23

                P + H + 8 = 2H + P
                    H + 8 = 2H
                        8 = H

Left Column: 2C + T + H = ? 2(10) + 6 + 8 = ? Therefore 34 = ?

Circles are 10 Triangles are 6 Pentagons are 8 Hearts are 7

This problem is based on 'Transportation problem' which is part of Operation Research. It is balanced traportation problem. Sum of summation of rows is 132 (I.e 40+32+31+29=132) . According to balanced transportation problem sum of summations of rows must be equal to sum of the summations of colums. So ?+35+31+32 must be equal to 132 Let unknown be x . Therefore x=132-(35+31+32).... X=34

Cramer's rule and Determinant

1
2
3
4
5

Blue circle = 40/4 = 10
Red triangule = 32 - 20 = 12/2 = 6

Third row = 6 + x + y + 10 = 31 - 16 = 15
Fourth row = 2x +y +6 = 29 - 6 = 23  

$\begin{vmatrix} x + y \\ x + 2y \end{vmatrix} = 1$

$\begin{vmatrix} x + 15 \\ x + 23 \end{vmatrix} = 8$

$\begin{vmatrix} 15 + y \\ 23 + 2y \end{vmatrix} = 7$

1
2
3
4

y = 8/1 = 8
x = 7/1 = 7

First column  = 10 + 10 + 6 + 8 = 34 

Its just like solving simultaneous equations...here variables are the symbols

I had to plug and chuck on the 3rd column until i had my answer key, circle = 10. Triangle = 6. Pentagon = 7. And heart with 8

Here,in first column-o+o+o+o=40 so, 4*o=40 →10=o. By this we get 6=^(triangle) in secound colmn .in 4th row o+^+&(heart)+#(polygon)=31.so,&=8and #=7

Hi I'm new

4C = 40 => C = 10

2C + 2T = 32 => T = 6

• T + P + H + B = 31 => P + H = 15

• 2H + P + T = 29 => P + 2H = 21

=> H = 8 & P = 7

In the first row: 4 circle=40, so circle=10 in the second row: 2 circle+2 triangle=32 so triangle=6 third row: triangle+pentagon+love+circle=31, so pentagon+love=31-16=15 Forth row : 2love+ pentagon +triangle=29, so 2love+pentagon=29-6=23 Now, 2love+pentagon-pentagon-love=23-15=8 so love=8 Now first column: 2 circle+triangle+love=20+6+8=34

Or as a simultaneous equation 2Hearts + 1Pentagon =29-6, less 1H +1P=31-10-6; H=8

The value of Circle is 10, Triangle is 6, pentagon is 7 and Heart is 8 respectively. If we add value of 2 Circle, 1 Triangle and 1 Heart is 34.

10 + 10 + 10 + 10 = 40 10 + 10 + 6 + 6 = 32 6 + 7 + 8 + 10 = 31

8 + 8 + 7 + 6 = 29

34 35 31 32

I know circles equal 10, I know triangles equal 6, I know pentagon + heart equal 15, Bottom row: 6 + 15 = 21, Second heart must equal 8,

No fancy math required……..

From Diagram ,value of Circle 10,value of triangle is 6 .With help of row 4 , we can conclude that value of Pentagon is 7 and that of Heart is 8 ,thus the answer is 10+10+6+8=34

40+32+31+29=35+32+31+a =>>a = 40+32+31+29 - 35 - 32 - 31 = 34

every circle = 10 triangle =6 heart = 8 hexagonal =7 so 10+10+6+8=34

First row All are circles so,40/4=10 ie circle=10 Second row 2circle 2triangle So,(32-20)/2=6 Similarly do for 3 and4 row and get answer (10+10+6+8)=34

No big deal!!! Just assume the shapes for variables and solve it simultaneously!!! Damn easy!!!

In the first row the sum is 40. Let 'a' represent the circles. Then 4a = 40 or a = 10. Let the triangles be 'b'. Then in the second row 2a +2b = 32 or b = 6. Let pentagon be 'x' and hearts be 'y'. In third row x+y=15 In last row x+2y=23 So y = 4 Hence in the first column 2a + b + y = 34

One can clearly see that there are circles, triangles, hearts, and pentagons. From this we can conclude that the pattern obviously comes from "bastos" which is a card game brought by the spaniards to the new world. The shapes symbolize the circle of life, the trangle is the food pyramid, the heart is obvious, and the pentagon where 7 virgins were sacrified every day for 7 years. So if one has an idea of the historical meaning of the shapes, one can conclude that the answer is 34.

Here's a solution that doesn't actually require solving for all variables:

From the first row 4C = 40, then C = 10.

Second row: 2C + 2T = 32 -> 20 +2T = 32 -> T = 6.

Third row: T + (P +H) + C = 31. Using substitution and algebra, (P + H) = 15.

Fourth row: H + (P + H) + T = 29, so H + 15 + 6 = 29, and P = 8.

We now have everything we need to add the first column; the value of the green pentagon doesn't matter. C + C + T + H = 34.

If you take 2nd column, you figure out the heart plus the Pentagon has to be 15. from the 2nd row you subtract 20 from 32 and end with 12. You 2 triangles equals 12 thus one triangle is 6.

Than if you take h + p = 15 p = 15 - h

2h + p + t = 29 2h + p = 29 - t 2h + p = 29 - 6 2h + p = 23 2h + 15 - h = 23 h = 8

2c + t + h = 2 * 10 + 6 + 8 = 20 + 6 + 8 = 34

C = Circle

T = Triangle

P = Pentagon

H = Heart

4C = 40 .... C = 10

2C + 2T = 32 .... 2T = 12 .... T = 6

P + T + H + C = 31 .... H + P= 15 (1)

2H + P + T = 29 .... From (1) .... H = 8

From C,T and H

Therefore.H + T + 2C = 8 + 6 + 20 = 34

? = 34