komfortabel Echo Fegen maths puzzles for class 4 Nationalismus ... - Free Printable

To solve this math puzzle, we need to find the missing numbers (the yellow diamonds) and the intermediate results (the blue squares). The grid works like a crossword puzzle for math: you can solve it by looking at the rows (horizontal) or the columns (vertical).

Let's break it down step-by-step. We will label the missing yellow diamonds as $D_1$ through $D_6$ and the blue squares as $S_1$ through $S_{12}$ to keep track, but we can just solve them logically.

Step 1: Solve Row 4
Look at the fourth row of numbers:
$20 - \text{Diamond} + 16 - 19 = 15$

Let's simplify the known numbers first:
$20 + 16 = 36$
$36 - 19 = 17$

So the equation becomes:
$17 - \text{Diamond} = 15$

To find the Diamond, we ask: What number subtracted from 17 equals 15?
$17 - 15 = 2$
So, the Diamond in Row 4 is 2.

Step 2: Solve Column 2
Now look at the second column (the vertical line starting with 18). We know the bottom number is a Diamond, and we just found some numbers above it. Let's look at the whole column equation from top to bottom.
The column reads:
$18 - \text{Blue Square}_1 + 24 + \text{Blue Square}_2 - \text{Diamond}_{Row4} - \text{Blue Square}_3 + 7 = \text{Bottom Diamond}$

This looks complicated because there are too many unknowns. Let's try a different column that is easier. Let's look at Column 5 (the one with 61, diamond, 58, 19, 37).

Actually, let's look at Row 3:
$16 + \text{Diamond} - \text{Diamond} + 58 = 66$
Wait, Row 3 has two diamonds. Let's call the first one $D_A$ and the second one $D_B$.
$16 + D_A - D_B + 58 = 66$
$74 + D_A - D_B = 66$
$D_A - D_B = 66 - 74$
$D_A - D_B = -8$
This means $D_B$ is 8 larger than $D_A$. This doesn't give us exact numbers yet. Let's hold on to this.

Let's look at Column 1 (far left):
$\text{Diamond}_1 - \text{Blue}_1 + 37 + \text{Blue}_2 + 16 - \text{Blue}_3 + 20 - \text{Blue}_4 = \text{Diamond}_5$
Too many unknowns.

Let's try Column 6 (far right):
$\text{Diamond}_2 - \text{Blue}_5 + 64 - \text{Blue}_6 + 66 - \text{Blue}_7 + 15 - \text{Blue}_8 = 78$
Also too many unknowns.

Let's go back to basics. We need to find a row or column with only one missing number.

We already solved Row 4's diamond ($D=2$).
Let's look at Column 3 (the middle vertical line):
It contains: $-\text{Diamond}_A$, $-\text{Blue}$, $-12$, $+\text{Blue}$, $-\text{Diamond}_B$, $-\text{Blue}$, $+16$, $-\text{Blue}$, $+27$.
This is messy.

Let's look at Row 2:
$37 + 24 - 12 + \text{Diamond} = 64$
Calculate the knowns:
$37 + 24 = 61$
$61 - 12 = 49$
So: $49 + \text{Diamond} = 64$
$\text{Diamond} = 64 - 49$
$\text{Diamond} = 15$
So, the Diamond in Row 2 is 15.

Now we have two diamonds found:
- Row 2, Col 5 Diamond = 15
- Row 4, Col 3 Diamond = 2

Let's update our knowledge.
Row 3 is: $16 + D_1 - D_2 + 58 = 66$.
We established $D_1 - D_2 = -8$, or $D_2 = D_1 + 8$.

Let's look at Column 5 again.
Top to bottom:
$61$ (Row 1)
$+$ (operator)
$\text{Blue Square}$ (Row 2 result area? No, the operators are between the numbers).

Let's trace Column 5 carefully.
The numbers in Column 5 are:
Row 1: 61
Row 2: Diamond (which we found is 15)
Row 3: 58
Row 4: 19
Row 5: 37

The operators in Column 5 are:
Between R1-R2: $+$
Between R2-R3: $-$
Between R3-R4: $-$
Between R4-R5: $+$

So the equation for Column 5 is:
$61 + 15 - 58 - 19 + 37 = \text{Final Result?}$
Wait, the last cell in Column 5 is part of the final answer row? No, the final row is separate.
Let's look at the structure.
The grid has 5 main rows of equations and 5 main columns of equations.
The "blue squares" are the intermediate answers for the columns? Or are they just placeholders?
Usually in these puzzles, the blue squares are the results of the vertical operations up to that point, or they connect the horizontal and vertical logic.

Let's look at Column 1:
Numbers: $\text{Diamond}_1$, (blank), 37, (blank), 16, (blank), 20, (blank), $\text{Diamond}_5$.
Operators: $+$, $-$, $+$, $+$, $-$, $-$, $=$, $+$
This interpretation is tricky. Let's look at the standard format.
Horizontal Rows:
R1: $D_1 + 18 - D_2 + 61 = D_3$
R2: $37 + 24 - 12 + D_4 = 64$ --> We found $D_4 = 15$.
R3: $16 + D_5 - D_6 + 58 = 66$
R4: $20 - D_7 + 16 - 19 = 15$ --> We found $D_7 = 2$.
R5: $D_8 + 7 + 27 + 37 = 78$

Let's solve Row 5:
$D_8 + 7 + 27 + 37 = 78$
Sum the knowns: $7 + 27 = 34$; $34 + 37 = 71$.
$D_8 + 71 = 78$
$D_8 = 78 - 71$
$D_8 = 7$
So, the Diamond in Row 5 (first box) is 7.

Now we have Diamonds:
$D_4$ (Row 2, pos 4) = 15
$D_7$ (Row 4, pos 2) = 2
$D_8$ (Row 5, pos 1) = 7

Let's look at the Vertical Columns now.
Column 1:
Starts with $D_1$.
Ops: $-$, $+$, $+$, $-$, $=$, $+$
Wait, the operators are in the cells between the numbers.
Col 1 Numbers: $D_1$, 37, 16, 20, $D_8$ (which is 7).
Let's check the operators in Col 1 vertically:
Row 1-2 gap: $-$
Row 2-3 gap: $+$
Row 3-4 gap: $+$
Row 4-5 gap: $-$
Row 5-6 gap: $=$
Row 6-7 gap: $+$ (This leads to the final answer?)

Actually, looking at the layout:
The blue squares are likely the results of the vertical calculations *at each step* or just the final result.
Let's look at Column 2:
Numbers: 18, (Blue), 24, (Blue), Diamond($D_5$), (Blue), Diamond($D_7=2$), (Blue), 7.
This seems overly complex.

Let's assume the Blue Squares are simply the result of the vertical operation ending at that row?
Or maybe the Blue Squares are just variables we don't need to find explicitly to get the Diamonds?
The question asks to "Find the missing numbers!" which usually refers to the Diamonds. The Blue Squares might be distractors or intermediate steps. However, sometimes the Blue Squares are required to solve other Diamonds.

Let's look at Column 5 again with the known Diamond $D_4=15$.
Col 5 Numbers: 61, $D_4(15)$, 58, 19, 37.
Vertical Operators:
Between 61 and 15: $+$
Between 15 and 58: $-$
Between 58 and 19: $-$
Between 19 and 37: $+$

Let's calculate the vertical sum for Column 5:
$61 + 15 = 76$
$76 - 58 = 18$
$18 - 19 = -1$
$-1 + 37 = 36$
Does this match anything? The last cell in Col 5 is a Blue Square, then below that is an "=" sign, then another Blue Square?
No, the last row is $D_8 + 7 + 27 + 37 = 78$.
The cell under 37 in Col 5 is a Blue Square. Then below that is "=". Then below that is nothing?
Actually, the grid ends with the row $D_8 + 7 + 27 + 37 = 78$.
The vertical columns seem to terminate at the blue squares in the row above the final equation?
Let's look at the very bottom row of operators.
Under Col 1: $=$
Under Col 2: $=$
Under Col 3: $=$
Under Col 4: $=$
Under Col 5: $=$
Under Col 6: $=$

And the final row is:
$\text{Diamond} + 7 + 27 + 37 = 78$
This implies the vertical columns result in the numbers used in the final horizontal equation?
No, the final horizontal equation has specific numbers: 7, 27, 37.
These numbers correspond to the columns 2, 3, 4, 5?
Col 2 has 18...
Col 3 has ...
Col 4 has ...
Col 5 has 61...

Let's re-read the grid structure.
There are 5 horizontal equations involving Diamonds.
There are 5 vertical equations involving Blue Squares?
Let's look at Column 2:
Top number: 18.
Operator: $-$
Next number: Blue Square? No, the Blue Square is in the intersection.
Usually, in these grids:
Horizontal: Number Op Number Op Number ... = Result
Vertical: Number Op Number Op Number ... = Result

Let's test if the Vertical Columns also equal something specific.
Look at Column 6 (Far Right).
Numbers: $D_3$ (Result of R1), $+$, $64$ (Result of R2), $-$, $66$ (Result of R3), $-$, $15$ (Result of R4), $=$, $78$ (Result of R5).
Wait, the last column contains the RESULTS of the horizontal rows.
R1 Result = $D_3$
R2 Result = 64
R3 Result = 66
R4 Result = 15
R5 Result = 78

So Column 6 is a vertical equation using the horizontal results!
Equation for Column 6:
$D_3 + 64 - 66 - 15 = 78$?
Let's check the operators in Column 6:
Between R1/R2: $+$
Between R2/R3: $-$
Between R3/R4: $-$
Between R4/R5: $=$ ?? No, the operator between R4 and R5 is blank/blue square?
Let's look at the image again.
Col 6 Operators:
Row 1-2: $+$
Row 2-3: $-$
Row 3-4: $-$
Row 4-5: $=$ (This is in the blue square row)
Row 5-6: (Empty/End)

Wait, the blue squares are in the rows *between* the number rows.
Let's map the vertical operators for Column 6 strictly:
1. Start with $D_3$ (Top Right Diamond)
2. Operator: $+$
3. Next Number: 64
4. Operator: $-$
5. Next Number: 66
6. Operator: $-$
7. Next Number: 15
8. Operator: $=$ (This suggests the result follows)
9. Result: 78

So, does $D_3 + 64 - 66 - 15 = 78$?
$D_3 + 64 - 66 - 15 = D_3 - 17$
$D_3 - 17 = 78$
$D_3 = 78 + 17$
$D_3 = 95$

Let's verify this. If $D_3 = 95$, let's plug it into Row 1.
Row 1: $D_1 + 18 - D_2 + 61 = D_3$
$D_1 + 18 - D_2 + 61 = 95$
$D_1 - D_2 + 79 = 95$
$D_1 - D_2 = 95 - 79$
$D_1 - D_2 = 16$

We now have a relationship between $D_1$ and $D_2$.

Now let's look at Column 1 (Far Left).
It uses the first number of each row.
Numbers: $D_1$, 37, 16, 20, $D_8$ (which is 7).
Operators in Col 1:
1. Start $D_1$
2. Op: $-$
3. Num: 37
4. Op: $+$
5. Num: 16
6. Op: $+$
7. Num: 20
8. Op: $=$
9. Result: $D_8$ (which is 7)

So, does $D_1 - 37 + 16 + 20 = 7$?
$D_1 - 37 + 36 = 7$
$D_1 - 1 = 7$
$D_1 = 8$

Great! We found $D_1 = 8$.

Now use the relationship $D_1 - D_2 = 16$:
$8 - D_2 = 16$
$D_2 = 8 - 16$
$D_2 = -8$

Negative numbers? In elementary math puzzles, negative numbers are rare but possible. Let's double check my reading of Column 1 operators.
Image check:
Col 1, Row 1-2 gap: Minus sign ($-$). Correct.
Col 1, Row 2-3 gap: Plus sign ($+$). Correct.
Col 1, Row 3-4 gap: Plus sign ($+$). Correct.
Col 1, Row 4-5 gap: Equals sign ($=$). Correct.
Result is $D_8$ (bottom left diamond).

Calculation:
$D_1 - 37 + 16 + 20 = D_8$
We found $D_8 = 7$ from Row 5 ($7+7+27+37=78$).
$D_1 - 37 + 36 = 7$
$D_1 - 1 = 7$
$D_1 = 8$. This seems solid.

Now check Row 1 again:
$D_1 + 18 - D_2 + 61 = D_3$
$8 + 18 - D_2 + 61 = D_3$
$87 - D_2 = D_3$

And Column 6 again:
$D_3 + 64 - 66 - 15 = 78$
$D_3 - 17 = 78$
$D_3 = 95$

Substitute $D_3$ back:
$87 - D_2 = 95$
$-D_2 = 95 - 87$
$-D_2 = 8$
$D_2 = -8$

Is it possible I misread an operator in Column 6?
Col 6 Ops:
R1-R2: $+$
R2-R3: $-$
R3-R4: $-$
R4-R5: $=$
Let's re-examine the image for Col 6 operators.
Between 61 (R1) and Diamond (R2)? No, Col 6 contains the RESULTS.
R1 Result: Diamond ($D_3$)
R2 Result: 64
R3 Result: 66
R4 Result: 15
R5 Result: 78

Ops between them:
Between $D_3$ and 64: The symbol is in the blue square. It looks like a Plus (+).
Between 64 and 66: The symbol is in the blue square. It looks like a Minus (-).
Between 66 and 15: The symbol is in the blue square. It looks like a Minus (-).
Between 15 and 78: The symbol is Equals (=).

So $D_3 + 64 - 66 - 15 = 78$ is correct based on visual inspection.
This leads to $D_2 = -8$.

Let's check if there's an alternative interpretation. Maybe the last operator is not minus?
Looking closely at the crop showing the right side...
Between 66 and 15, there is a minus sign.
Between 64 and 66, there is a minus sign.
Between $D_3$ and 64, there is a plus sign.

Okay, let's look at Column 5 to see if it helps verify or if I missed a constraint.
Col 5 Numbers: 61, $D_4$(15), 58, 19, 37.
Ops:
61 to 15: $+$
15 to 58: $-$
58 to 19: $-$
19 to 37: $+$
Result: Blue Square in Row 6?
The blue square in Col 5, Row 6 (above the final row) would be the result of this vertical chain.
$61 + 15 - 58 - 19 + 37 = ?$
$76 - 58 = 18$
$18 - 19 = -1$
$-1 + 37 = 36$
So the blue square above the 37 in the final row is 36?
But the final row has the number 37 in that position.
Wait, the final row is:
$D_8 + 7 + 27 + 37 = 78$
The numbers 7, 27, 37 are printed. They are not results of the columns above them necessarily. They are just part of the final equation.

However, often in these puzzles, the "Blue Squares" are just intermediate values that ensure the vertical and horizontal consistency. If the puzzle is consistent, my derived values should work for all columns.

Let's find the remaining Diamonds $D_5$ and $D_6$ from Row 3.
Row 3: $16 + D_5 - D_6 + 58 = 66$
$74 + D_5 - D_6 = 66$
$D_5 - D_6 = -8$ => $D_6 = D_5 + 8$

We need another equation linking $D_5$ and $D_6$.
$D_5$ is in Column 3.
$D_6$ is in Column 4.

Let's analyze Column 3:
Numbers: 18 (from R1? No, Col 3 is the third column).
Let's identify the columns properly.
Col 1: $D_1$, 37, 16, 20, $D_8$
Col 2: 18, Blue, 24, Blue, $D_5$, Blue, $D_7(2)$, Blue, 7
Wait, Row 1 is: $D_1 + 18 - D_2 + 61 = D_3$
So:
Col 1: $D_1$
Col 2: 18
Col 3: $D_2$ <-- Wait, in Row 1, the third term is $- D_2$. So $D_2$ is in Column 3.
Col 4: 61
Col 5: $D_3$ (Result) -- Wait, earlier I said Col 6 is results. Let's recount columns.

Grid Layout:
Cell 1: $D_1$ (Diamond)
Cell 2: $+$
Cell 3: 18
Cell 4: $-$
Cell 5: $D_2$ (Diamond)
Cell 6: $+$
Cell 7: 61
Cell 8: $=$
Cell 9: $D_3$ (Diamond)

So there are 9 columns in the grid structure?
No, the operators are in their own columns?
If operators are in columns, then:
Col 1: $D_1$
Col 2: $+$
Col 3: 18
Col 4: $-$
Col 5: $D_2$
Col 6: $+$
Col 7: 61
Col 8: $=$
Col 9: $D_3$

If this is the case, the "Vertical Equations" run down these specific columns.
Let's check the vertical alignment.
Column 1 contains: $D_1$, $-$, 37, $+$, 16, $+$, 20, $=$, $D_8$.
This matches my previous calculation for Col 1.
$D_1 - 37 + 16 + 20 = D_8$.
Result: $D_1 = 8, D_8 = 7$.

Column 3 contains: 18, $-$, 24, $+$, $D_5$, $-$, $D_7(2)$, $=$, 7.
Let's verify the operators in Column 3 (which holds the number 18 in Row 1).
Row 1: 18
Op R1-R2: $-$ (Visible in blue square)
Row 2: 24
Op R2-R3: $+$ (Visible in blue square)
Row 3: $D_5$ (Diamond)
Op R3-R4: $-$ (Visible in blue square)
Row 4: $D_7$ (which is 2)
Op R4-R5: $=$ (Visible in blue square)
Row 5: 7

Equation for Column 3:
$18 - 24 + D_5 - 2 = 7$
$-6 + D_5 - 2 = 7$
$D_5 - 8 = 7$
$D_5 = 15$

So, $D_5 = 15$.

Now we can find $D_6$ using the Row 3 relation:
$D_6 = D_5 + 8$
$D_6 = 15 + 8$
$D_6 = 23$

Let's verify $D_6$ with Column 5?
Wait, where is $D_6$ located?
Row 1: $D_1$ (Col 1), 18 (Col 3), $D_2$ (Col 5), 61 (Col 7), $D_3$ (Col 9).
Row 2: 37 (Col 1), 24 (Col 3), 12 (Col 5?? No).

Let's re-map Row 2:
$37 + 24 - 12 + D_4 = 64$
Cells:
Col 1: 37
Col 2: $+$
Col 3: 24
Col 4: $-$
Col 5: 12
Col 6: $+$
Col 7: $D_4$
Col 8: $=$
Col 9: 64

So $D_4$ is in Column 7.
And 12 is in Column 5.

Let's re-map Row 3:
$16 + D_5 - D_6 + 58 = 66$
Col 1: 16
Col 2: $+$
Col 3: $D_5$
Col 4: $-$
Col 5: $D_6$
Col