Question

In the figure, the number in any cell is obtained by adding the two numbers in the cells directly below it. For example, the 9 marked in the figure is obtained by adding the 4 and 5 directly below it.

The value of X - Y is

• A. 2
• B. 3
• C. 4
• D. 5

Hint:

Write the row above the base first using the addition rule, then match the cell the figure labels "Y + 29" with the same cell built from the base row using X.

Answer 1

The Correct Option is C

Step 1: See the pyramid as weighted sums of the base row.
In an addition pyramid like this, any cell can be written directly as a weighted sum of the base cells sitting under it, using the same numbers that appear in Pascal's triangle. Climbing four levels above the base always folds in the row 1, 4, 6, 4, 1.

Step 2: Write the apex using this weighting.
The apex sits four rows above the base row $Y, 4, 5, 2, X$, so:
\[ \text{apex} = 1(YY) + 4(44) + 6(55) + 4(22) + 1(XX) = Y + 16 + 30 + 8 + X = X + Y + 54 \]
Since the apex is given as 68:
\[ X + Y + 54 = 68 \implies X + Y = 14 \]

Step 3: Write the labeled cell using the same idea.
The cell the figure marks "Y + 29" sits three rows above the base and covers the last four base cells (4, 5, 2, X), climbing three levels, which folds in Pascal's row 1, 3, 3, 1:
\[ 1(44) + 3(55) + 3(22) + 1(XX) = 4 + 15 + 6 + X = X + 25 \]
The figure tells us this same cell equals $Y + 29$, so:
\[ X + 25 = Y + 29 \implies X - Y = 4 \]

Step 4: Solve the pair of equations for X and Y.
We now have two equations: $X + Y = 14$ and $X - Y = 4$. Adding them gives $2X = 18$, so $X = 9$. Substituting back, $9 + Y = 14$, so $Y = 5$.

Step 5: Verify.
Putting $X = 9, Y = 5$ into the base row gives 5, 4, 5, 2, 9. Climbing up: next row is 9, 9, 7, 11; next row is 18, 16, 18; next row is 34, 34; and the apex is $34 + 34 = 68$, matching the given value exactly.

Final Answer:
$X - Y = 4$, option (CC).\[ \boxed{X-Y=4} \]