Question 8

The number in each block is the sum of the numbers in the two blocks directly below it.

Step 1 — Solving Pyramid (i)

Let us label the empty blocks with variables. The top block is 50. The block below it on the right is 22. Let the block below 50 on the left be $a$.

$a + 22 = 50$

We can find the value of $a$.

$a = 50 - 22$

$\boxed{a = 28}$

Now we look at the third row from the top. Let the leftmost block be $b$. Let the middle block be $c$. Let the rightmost block be $d$. We know $a = b + c$.

$28 = b + c$

We also know that the block 22 is the sum of $c$ and $d$.

$c + d = 22$

Now we look at the bottom row. The leftmost block is 4. Let the second block be $e$. The third block is 6. Let the fourth block be $f$. We can write equations for $b$ and $c$ in terms of $e$.

$b = 4 + e$

$c = e + 6$

Let us substitute these into the equation $28 = b + c$.

$28 = (4 + e) + (e + 6)$

$28 = 10 + 2e$

$2e = 28 - 10$

$2e = 18$

$\boxed{e = 9}$

Now that we know $e$, we can find $b$ and $c$.

$b = 4 + 9$

$\boxed{b = 13}$

$c = 9 + 6$

$\boxed{c = 15}$

With $c = 15$, we can find $d$ using $c + d = 22$.

$15 + d = 22$

$d = 22 - 15$

$\boxed{d = 7}$

Finally, we find $f$ using the relationship $d = 6 + f$.

$7 = 6 + f$

$f = 7 - 6$

$\boxed{f = 1}$

Step 2 — Solving Pyramid (ii)

Let us label the empty blocks with variables. The top block is $a$. The block below it on the left is 40. Let the block below it on the right be $b$. In the third row, let the leftmost block be $c$. Let the middle block be $d$. The rightmost block is 9. In the bottom row, the leftmost block is 5. Let the second block be $e$. The third block is 7. Let the fourth block be $f$.

We can start from the right side of the bottom two rows. The block 9 is the sum of the two blocks below it.

$7 + f = 9$

We find $f$.

$f = 9 - 7$

$\boxed{f = 2}$

Now we use the relationship for the block 40. The block 40 is the sum of $c$ and $d$.

$c + d = 40$

We can express $c$ and $d$ in terms of $e$.

$c = 5 + e$

$d = e + 7$

Substitute these into $c + d = 40$.

$(5 + e) + (e + 7) = 40$

$12 + 2e = 40$

$2e = 40 - 12$

$2e = 28$

$\boxed{e = 14}$

Now that we know $e$, we can find $c$ and $d$.

$c = 5 + 14$

$\boxed{c = 19}$

$d = 14 + 7$

$\boxed{d = 21}$

Now we find $b$. The block $b$ is the sum of $d$ and 9.

$b = d + 9$

$b = 21 + 9$

$\boxed{b = 30}$

Finally, we find $a$. The top block $a$ is the sum of 40 and $b$.

$a = 40 + b$

$a = 40 + 30$

$\boxed{a = 70}$

Step 3 — Solving Pyramid (iii)

Let us label the empty blocks with variables. The top block is 35. Let the block below it on the left be $a$. Let the block below it on the right be $b$. In the third row, let the leftmost block be $c$. Let the middle block be $d$. The rightmost block is 7. In the bottom row, the leftmost block is 3. The second block is 5. Let the third block be $e$. Let the fourth block be $f$.

We can start from the bottom left. The block $c$ is the sum of 3 and 5.

$c = 3 + 5$

$\boxed{c = 8}$

Now we use the relationship for the top block 35. The block 35 is the sum of $a$ and $b$.

$a + b = 35$

We can express $a$ and $b$ in terms of $c$ and $d$. The block $a$ is the sum of $c$ and $d$.

$a = c + d$

The block $b$ is the sum of $d$ and 7.

$b = d + 7$

Substitute these into $a + b = 35$.

$(c + d) + (d + 7) = 35$

$c + 2d + 7 = 35$

We know $c = 8$.

$8 + 2d + 7 = 35$

$15 + 2d = 35$

$2d = 35 - 15$

$2d = 20$

$\boxed{d = 10}$

Now that we know $d$, we can find $e$. The block $d$ is the sum of 5 and $e$.

$5 + e = d$

$5 + e = 10$

$e = 10 - 5$

$\boxed{e = 5}$

Now that we know $e$, we can find $f$. The block 7 is the sum of $e$ and $f$.

$e + f = 7$

$5 + f = 7$

$f = 7 - 5$

$\boxed{f = 2}$

Now we find $a$ using $a = c + d$.

$a = 8 + 10$

$\boxed{a = 18}$

Finally, we find $b$ using $b = d + 7$.

$b = 10 + 7$

$\boxed{b = 17}$

Answer

(i) The missing numbers are $a=28, b=13, c=15, d=7, e=9, f=1$. (ii) The missing numbers are $a=70, b=30, c=19, d=21, e=14, f=2$. (iii) The missing numbers are $a=18, b=17, c=8, d=10, e=5, f=2$.