Question

In the arrangement below, each number is the non-negative difference of the two numbers above it. What is the sum of the four greatest distinct possible values for $z$? [asy] unitsize(1 cm); label("$40$", (0,0)); label("$4$", (1,0)); label("$18$", (2,0)); label("$48$", (3,0)); label("$40$", (4,0)); label("$76$", (5,0)); label("$z$", (6,0)); label("$36$", (0.5,-0.5)); label("$14$", (1.5,-0.5)); label("$30$", (2.5,-0.5)); label("$8$", (3.5,-0.5)); label("$36$", (4.5,-0.5)); label("$y$", (5.5,-0.5)); label("$22$", (1,-1)); label("$16$", (2,-1)); label("$22$", (3,-1)); label("$28$", (4,-1)); label("$x$", (5,-1)); label("$6$", (1.5,-1.5)); label("$6$", (2.5,-1.5)); label("$6$", (3.5,-1.5)); label("$w$", (4.5,-1.5)); label("$0$", (2,-2)); label("$0$", (3,-2)); label("$0$", (4,-2)); [/asy]

Answer 1

The sum of the four greatest distinct possible values for z is 148.

Let's work our way from the bottom to the top, calculating each value based on the difference of the two numbers above it:

w = x - 28

x = y + 22

y = z - 76

Now, substituting these expressions back into the equations:

w = (y + 22) - 28 = y - 6

x = (z - 76) + 22 = z - 54

y = z - 76

Now, we use these values to express the top row in terms of z:

z = (z - 54) - 36 = z - 90

Solving for z, we find z = 90.

Now, we can substitute this value into the expressions we found earlier:

y = 90 - 76 = 14

x = 90 - 54 = 36

w = 14 - 6 = 8

Finally, we can find the four greatest distinct possible values for z: 90, 36, 14, and 8. Their sum is 90 + 36 + 14 + 8 = 148.

The question probable may be:

What is the sum of the four greatest distinct possible values for z in the given arrangement, where each number is the non-negative difference of the two numbers above it?

w = x - 28

x = y + 22

y = z - 76