The Other Side of Zero · Lesson 8 of 10
Integer Patterns and Puzzles
“Investigate grids, dice, sequences, cards, and token strings using signed-number reasoning.”
• Complete hollow grids with a common border sum. • Explain why some puzzles admit several solutions. • Test and explain fixed totals in row-and-column selection grids. • List possible and impossible sums systematically. • Continue integer sequences using a stated pattern. • Determine possible result signs, including zero.
Four borders, one common sum
A hollow integer grid is a 3 × 3 square whose centre is unused. Add the three entries along its top, bottom, left, and right borders. When all four totals agree, their common value is the border sum. Corners belong to two borders, so changing a corner affects two totals.
Problem
Verify the four border sums of the first grid.
- 1.Top: 4 + (−1) + (−3) = 0. Bottom: −1 + (−1) + 2 = 0.
- 2.Left: 4 + (−3) + (−1) = 0. Right: −3 + 1 + 2 = 0.
- 3.All four agree, so the border sum is 0. The fact that one border works alone would not be enough.
Problem
Find the border sum for [[5, −3, −5], [0, empty, −5], [−8, −2, 7]].
- 1.Top: 5 − 3 − 5 = −3; bottom: −8 − 2 + 7 = −3.
- 2.Left: 5 + 0 − 8 = −3; right: −5 − 5 + 7 = −3.
- 3.Every border has sum −3. A common total can be negative, zero, or positive.
Finding missing entries
Use a complete border, or a border with only one blank, to find an unknown entry. If a border must total −2 and already contains 6 and 8, the missing entry is −2 − (6 + 8) = −16. Check each border after filling the grid; a new corner is shared by two conditions.
Problem
Complete a grid with top row 6, 8, blank; middle row blank, empty, −5; bottom row blank, −2, blank. Its border sum must be −2.
- 1.The top-right entry is −2 − 6 − 8 = −16.
- 2.The right border is −16 + (−5) + ? = −2, giving bottom-right +19.
- 3.The bottom border is ? − 2 + 19 = −2, giving bottom-left −19.
- 4.The left border is 6 + ? − 19 = −2, giving middle-left +11. Each new entry was forced by an already determined border.
Problem
Only top-left 7 and right-middle −5 are fixed. Find two grids with border sum −4.
- 1.One solution is [[7, −2, −9], [−3, empty, −5], [−8, −6, 10]]. Its borders all total −4.
- 2.Another is [[7, −3, −8], [−3, empty, −5], [−8, −5, 9]]. Its borders also all total −4.
- 3.The extra freedom allows a different corner choice with compensating changes on adjoining borders. Different completed pictures can satisfy the same conditions.
A puzzle with fewer given entries can leave choices available. To build your own puzzle, first create a valid completed grid, then remove selected entries. Solve the puzzle yourself before challenging a classmate. Decide whether one solution is intended or whether several valid solutions should be accepted.
Selecting one entry from each row and column
In another puzzle, circle an entry and cross out its whole row and column. Repeat with an uncrossed entry until every row and column has been used. In a 4 × 4 grid, the four circled entries occupy four different rows and four different columns. Add those four values.
Problem
Compare selections (−1, 9, −7, −2) and (3, −1, −2, −1) in the grid.
- 1.The first can use row 2 column 2, row 1 column 4, row 4 column 1, and row 3 column 3.
- 2.Its total is −1 + 9 − 7 − 2 = −1.
- 3.The diagonal selection uses one entry from each row and each column too. Its total is 3 − 1 − 2 − 1 = −1.
- 4.Equal answers from two trials suggest a pattern, but explaining the arrangement tells us why every allowed selection works.
Compare each row with the top row. Row 2 subtracts 5 from every top-row entry, row 3 subtracts 2, and row 4 subtracts 10. Every allowed selection uses each column once, so its top-row parts add to 3 + 4 + 0 + 9 = 16. It also uses each row once, so its row changes add to 0 − 5 − 2 − 10 = −17. The total must therefore be 16 + (−17) = −1, whatever the selection.
| Row | Change from the top row |
|---|---|
| 1 | 0 |
| 2 | −5 in every column |
| 3 | −2 in every column |
| 4 | −10 in every column |
Problem
Does every selection have the same total in [[7, 10, 13, 16], [−2, 1, 4, 7], [−11, −8, −5, −2], [−20, −7, −14, −11]]?
- 1.The diagonal gives 7 + 1 − 5 − 11 = −8.
- 2.Another allowed choice is 7, 4, −2, −7, using columns 1, 3, 4, 2 respectively. Its total is 7 + 4 − 2 − 7 = +2.
- 3.The totals differ, so a fixed-total claim does not hold for this arrangement.
- 4.The last row follows a −27 change from the top row in all but its second entry. Replacing −7 with −17 restores that pattern. Then every selection totals 7 + 10 + 13 + 16 − 9 − 18 − 27 = −8.
A second consistently arranged grid is [[−11, −10, −9, −8], [−7, −6, −5, −4], [−3, −2, −1, 0], [1, 2, 3, 4]]. Its row changes are 0, +4, +8, +12. Its top-row total is −38 and the changes total +24, so every allowed selection gives −14. You can create new fixed-total grids by using the same change across every column of a row.
Several equal trial results suggest a pattern. A row-by-row explanation establishes why every permitted choice works. One counterexample, however, is enough to disprove a claim that every choice has the same result.
Which dice sums are possible?
Two dice each have faces −1, 2, −3, 4, −5, 6. Their smallest possible sum is −10, using −5 twice, and their largest is +12, using 6 twice. These limits do not mean that every integer in between can occur. A systematic table checks all pairs without depending on guesses.
| First / second die | −1 | +2 | −3 | +4 | −5 | +6 |
|---|---|---|---|---|---|---|
| −1 | −2 | +1 | −4 | +3 | −6 | +5 |
| +2 | +1 | +4 | −1 | +6 | −3 | +8 |
| −3 | −4 | −1 | −6 | +1 | −8 | +3 |
| +4 | +3 | +6 | +1 | +8 | −1 | +10 |
| −5 | −6 | −3 | −8 | −1 | −10 | +1 |
| +6 | +5 | +8 | +3 | +10 | +1 | +12 |
Problem
List the impossible totals between −10 and +12 inclusive.
- 1.Read the pair table and collect each distinct result. The possible totals are −10, −8, −6, −4, −3, −2, −1, 1, 3, 4, 5, 6, 8, 10, 12.
- 2.Compare this list with every integer from −10 to +12.
- 3.The missing totals are −9, −7, −5, 0, 2, 7, 9, and 11. For example, 6 needs −6 to make zero, but −6 is not a face on these dice.
Sequences, cards, and repeated groups
To continue a sequence, first state the pattern you are using. A finite list can fit more than one pattern, so your reasoning is part of the answer. Integers allow sequences to continue naturally through zero instead of stopping there.
Problem
Continue the chapter’s three sequences.
- 1.For −40, −34, −28, −22, add 6 each time: −16, −10, −4 follow.
- 2.For 3, 4, 2, 5, 1, 6, 0, 7, alternate a falling sequence 3, 2, 1, 0, −1, −2 with a rising sequence 4, 5, 6, 7, 8, 9. The next three terms are −1, 8, −2.
- 3.For blank, blank, 12, 6, 1, −3, −6, blank, blank, blank, use changes −8, −7, −6, −5, −4, −3, −2, −1, 0. The terms become 27, 19, 12, 6, 1, −3, −6, −8, −9, −9.
Problem
Choose from +1, +7, +18, −5, −2, −9 to make an expression with value −30.
- 1.Use the cards −2, −9, +18, and +1 once each.
- 2.The expression is (−2) + (−9) − (+18) − (+1).
- 3.Combine in stages: −2 − 9 = −11; −11 − 18 = −29; −29 − 1 = −30.
- 4.A card’s own sign stays with it; the operation before the card still determines whether it is added or subtracted.
Problem
A string repeats +, +, +, −, − until it contains 100 tokens. What is its value?
- 1.One group of five tokens has value 3 − 2 = +1.
- 2.There are 100 ÷ 5 = 20 complete groups and no leftover tokens.
- 3.Each group contributes +1, so the whole string has value +20. There are 60 positives and 40 negatives, which gives the same result.
Can you predict the result’s sign?
Some combinations always have the same sign, while others depend on the sizes. Two negative addends always have a negative sum. A positive minus a negative is always positive. But opposite-sign addition can be positive, negative, or zero; equal sizes cancel.
| Combination of nonzero integers | Possible result signs | Examples |
|---|---|---|
| positive + negative | positive, zero, or negative | 5 + (−2) = 3; 2 + (−2) = 0; 2 + (−5) = −3 |
| negative + positive | positive, zero, or negative | −2 + 5 = 3; −2 + 2 = 0; −5 + 2 = −3 |
| negative + negative | negative | −2 + (−3) = −5 |
| positive − negative | positive | 2 − (−3) = 5 |
| negative − positive | negative | −2 − 3 = −5 |
| negative − negative | positive, zero, or negative | −2 − (−5) = 3; −2 − (−2) = 0; −5 − (−2) = −3 |
| positive − positive | positive, zero, or negative | 5 − 2 = 3; 2 − 2 = 0; 2 − 5 = −3 |
Quiz
What is the border sum of [[5, −3, −5], [0, empty, −5], [−8, −2, 7]]?
What does an allowed selection in a 4 × 4 row-and-column puzzle use?
Which total is impossible with the signed dice −1, 2, −3, 4, −5, 6?
What follows −40, −34, −28, −22 when the change is +6?
What is the value of 100 tokens repeating +,+,+,−,−?
Which outcomes can a positive integer plus a negative integer have?
Practice Problems
- Complete a hollow grid with top-left −10, right-middle −5, bottom-left 9, and border sum +4.
- Give another solution for the grid with top-left 7, right-middle −5, and border sum −4.
- Explain why the grid with top row 6, 8, blank, right-middle −5, bottom-middle −2, and border sum −2 is uniquely determined.
- Play the first 4 × 4 selection game with three different choices, then explain why the sum is always −1.
- Show two choices in the grid containing −7 that give different totals. Explain how replacing it with −17 changes the pattern.
- Construct a 3 × 3 row-and-column selection grid with a fixed total. State how you made it.
- Use the signed-dice table to find one way each to obtain −3, +1, and +10. Explain why +2 is impossible.
- Continue each of the three sequence patterns and state its rule.
- Make −30 using the integer cards and check your result.
- Give examples of negative minus negative with a positive, zero, and negative result.
- Find the value of 103 tokens with the same +,+,+,−,− pattern.
One solution is [[−10, 10, 4], [5, empty, −5], [9, −10, 5]]. Every border totals +4.
Key Takeaways
• A border-grid answer must satisfy all four borders. • Available choices can give several valid solutions. • Explain a fixed total through the arrangement, not just repeated trials. • Minimum and maximum dice totals do not guarantee every intermediate total. • State the pattern you use when continuing a sequence. • Opposite-sign addition and same-sign subtraction can give zero as well as either sign.