1
Recover a value in a magic square
Not answeredIn the square below, the three numbers in every row, every column, and both diagonals have the same sum. Find x.
x
8
10
1
x
Lesson 14.7 · Test 14
Chapter 14 Exercise
Number-filling arrays, magic squares, neighbor rules, and number networks. Every diagram and instruction needed to complete the test is included on this page.
Exercise directions
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- Review and improve. Use the help available for this submission. Edit the original answer fields, then submit again.
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Tips for this exercise
Use each diagram directlyEnter values in the provided grids or circles. Numbers may be decimals, fractions, or negative unless a question restricts them. You never need the scanned this lesson.
Show enough structureFor construction questions, complete every requested cell or node, not only the final number.
Number-Filling Arrays
Complete all twelve questions. Each question is worth 10 points.
120 points
2
Find the common line sum
Not answeredIn a 3×3 magic square, the sums of all three rows, all three columns, and both diagonals are the same number A. The three shown entries are fixed. Find A.
2010
1
10
A
Optional scratch work
3
Count entries below 1
Not answeredFill a 3×3 grid with nine different numbers so that every row, every column, and both diagonals have sum 3. Among the nine entries, how many must be less than 1?
Optional working construction
1,1
1,2
1,3
2,1
2,2
2,3
3,1
3,2
3,3
This grid is optional; the requested count below is required.
entries
Optional explanation
4
Use equal row, column, and diagonal sums
Not answeredImagine completing the blank cells so that every row, every column, and both diagonals have the same sum. Only the value of x is required below; use the scratch space for a completed grid if helpful. Find x.
x
2
37
16
x
Optional scratch work
5
Construct a magic square with sum 27
Not answeredIn the 3×3 square, place 5 in Row 1, Column 3 and 6 in Row 2, Column 1. Fill every remaining cell so that all rows, columns, and diagonals have sum 27.
Your complete square
1,1
1,2
5
6
2,2
2,3
3,1
3,2
3,3
Optional scratch work or verification notes
6
Use one-digit entries
Not answeredEach cell contains a digit from 0 through 9. Repeated digits are allowed; they do not affect the requested value. Every row, every column, and both diagonals have the same sum. Find x.
9
x
1
x
Optional scratch work
7
Keep each side sum at 5 while changing the total
Not answeredIn the model, the center cell is not used. The three numbers along each outside side sum to 5, and all eight outside numbers total 12. Rearrange or choose any numbers to complete two new border grids: one with total 13 and one with total 14. You may choose new numbers and repeat values; you do not have to reuse the model’s numbers. In both grids, every outside side must still sum to 5.
2
1
2
1
1
2
1
2
Total 13
Total 14
Optional verification notes
8
Investigate an inconsistent magic square
Not answeredIn the nine-cell square, every row, every column, and both diagonals have the same sum. Explain why no value of N can satisfy all the givens.
original correction: The original book contains inconsistent givens in this question. The grid below preserves them. Your task is to diagnose the contradiction, rather than find a number. Compare the sum forced by the first column with the sum forced by the opposite corners and the center.
8
N
6
15
12
N
Explain the contradiction (required)
9
Combine a common horizontal difference and vertical quotient
Not answeredWithin each row, the difference between neighboring entries is constant (right number minus left number). Different rows may have different differences. Within each column, divide each entry by the one immediately above it: all three quotients in that column must agree. Do not assume in advance that different columns have the same quotient. Find a+b×c.
4
8
c
18
b
a
162
a+b×c
Optional scratch work
10
Use three side sums on a triangle
Not answeredSuppose six different numbers fill the circles. A, B, C are the three side-middle values, not the corner values. The three numbers on each side of the triangle sum to 12. If A+B+C=18, what is the sum of the three corner values? Only their sum is required, not a full construction.
sum
Optional scratch work
11
Fill a network so its seven edge differences are 1 through 7
Not answeredPlace the numbers 1,2,3,4,5,6,7,8, each exactly once, in the eight circles. For every drawn segment, subtract the smaller endpoint value from the larger endpoint value. The seven differences must be exactly 1,2,3,4,5,6,7, each once.
On a small screen, scroll the diagram sideways to reach every circle.
A
C
F
H
B
D
E
G
Optional list of your seven edge differences
12
Maximize the sum of twelve adjacent differences
Not answeredPlace the digits 1, 2, …, 9, each exactly once, in a 3×3 grid. For every horizontally or vertically adjacent pair, calculate larger minus smaller. There are twelve such differences. Arrange the digits so that the sum of the twelve differences is as large as possible. Give both your arrangement and the maximum sum.
Your maximizing arrangement
1,1
1,2
1,3
2,1
2,2
2,3
3,1
3,2
3,3
maximum
Explain why your value is maximal (required)
Complete saved responses and working
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