Question

7. Consider a topograph with values 1,7,-2 as in the margin (pictured).

Fill in the ?s to satisfy the arithmetic progression rule
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Answer 1

see below

Explanation:

The "arithmetic progression rule" requires the numbers on either side of an edge make an arithmetic progression with the numbers at either end.

If we label the variables 'a', 'b', 'c' clockwise from top, then the rule means we have ...

2a -b +2c = 1

2a +2b -c = 7

-a +2b +2c = -2

Solution

Adding twice the second equation to each of the other two gives ...

2(2a +2b -c) +(2a -b +2c) = 2(7) +(1)

6a +3b = 15 . . . . [eq4]

and

2(2a +2b -c) +(-a +2b +2c) = 2(7) +(-2)

3a +6b = 12 . . . . [eq5]

Subtracting [eq5] from twice [eq4] we have ...

2(6a +3b) -(3a +6b) = 2(15) -(12)

9a = 18

a = 2

From [eq4], we can find b:

b = (15 -6a)/3 = 5 -2a = 5 -2(2) = 1

From [eq2] we can find c:

c = 2(a+b) -7 = 2(2+1) -7 = -1

These values are shown on the diagram below.