Explorations with Integers
A hollow integer grid
There is something special about the numbers in these two grids. Let us explore what that is.
| 4 | –1 | –3 |
|---|---|---|
| –3 | 1 | |
| –1 | –1 | 2 |
| 5 | –3 | –5 |
|---|---|---|
| 0 | –5 | |
| –8 | –2 | 7 |
| Top row: | 4 + (–1) + (–3) = 0 | 5 + (– 3) + (– 5) = |
|---|---|---|
| Bottom row: | (– 1) + (– 1) + 2 = 0 | (– 8) + (– 2) + 7 = |
| Left column: | 4 + (– 3) + (– 1) = 0 | 5 + 0 + (– 8) = |
| Right column: | (– 3) + 1 + 2 = 0 | (– 5) + (– 5) + 7 = |
In each grid, the numbers in each of the two rows (the top row and the bottom row) and the numbers in each of the two columns (the leftmost column and the rightmost column) add up to give the same number. We shall call this sum as the 'border sum'. The border sum of the first grid is '0'.
Figure it Out
1. Do the calculations for the second grid above and find the border sum.
2. Complete the grids to make the required border sum:
| –10 | ||
|---|---|---|
| –5 | ||
| 9 |
Border sum is +4
| 6 | 8 | |
|---|---|---|
| –5 | ||
| –2 |
Border sum is –2
| 7 | ||
|---|---|---|
| –5 | ||
Border sum is –4
3. For the last grid above, find more than one way of filling the numbers to get border sum –4.
4. Which other grids can be filled in multiple ways? What could be the reason?
5. Make a border integer square puzzle and challenge your classmates.
An amazing grid of numbers!
Below is a grid having some numbers. Follow the steps as shown until no number is left.
| 3 | 4 | 0 | 9 |
|---|---|---|---|
| –2 | –1 | –5 | 4 |
| 1 | 2 | –2 | 7 |
| –7 | –6 | –10 | –1 |
Circle any number Strike out the row and column of the chosen number Circle any unstruck number
When there are no more unstruck numbers, STOP. Add the circled numbers.
In the example below, the circled numbers are – 1, 9, –7, –2. If you add them, you get
Figure it Out
1. Try afresh, choose different numbers this time. What sum did you get? Was it different from the first time? Try a few more times!
2. Play the same game with the grids below. What answer did you get?
| 7 | 10 | 13 | 16 |
|---|---|---|---|
| –2 | 1 | 4 | 7 |
| –11 | –8 | –5 | –2 |
| –20 | –7 | –14 | –11 |
| –11 | –10 | –9 | –8 |
|---|---|---|---|
| –7 | –6 | –5 | –4 |
| –3 | –2 | –1 | 0 |
| 1 | 2 | 3 | 4 |
3. What could be so special about these grids? Is the magic in the numbers or the way they are arranged or both? Can you make more such grids?
Figure it Out
1. Write all the integers between the given pairs, in increasing order.
a. 0 and –7:
b. –4 and 4:
c. –8 and –15:
d. –30 and –23:
2. Give three numbers such that their sum is –8. [[]]
3. There are two dice whose faces have these numbers: –1, 2, –3, 4, –5, 6. The smallest possible sum upon rolling these dice is –10 = (–5) + (–5) and the largest possible sum is 12 = (6)+(6). Some numbers between (–10) and (+12) are not possible to get by adding numbers on these two dice. Find those numbers.
4. Solve these:
| Expression | Answer |
|---|---|
| 8 – 13 | |
| (– 8) – (13) | |
| (– 13) – (– 8) | |
| (– 13) + (– 8) | |
| 8 + (– 13) | |
| (– 8) – (– 13) | |
| (13) – 8 | |
| 13 – (– 8) |
5. Find the years below.
a. From the present year, which year was it 150 years ago?
b. From the present year, which year was it 2200 years ago?
Hint: Recall that there was no year 0.
c. What will be the year 320 years after 680 BCE?
6. Complete the following sequences:
a. (–40), (–34), (–28), (–22),
b. 3, 4, 2, 5, 1, 6, 0, 7,
c.
7. Here are six integer cards: (+1), (+7), (+18), (–5), (–2), (–9). You can pick any of these and make an expression using addition(s) and subtraction(s).
Here is an expression: (+18)+(+1)–(+7) – (–2) which gives a value (+14). Now, pick cards and make an expression such that its value is closer to (– 30).
8. The sum of two positive integers is always positive but a (positive integer) – (positive integer) can be positive or negative. What about
a. (positive) – (negative): Always
b. (positive) + (negative): Can be
c. (negative) + (negative): Always
d. (negative) – (negative): Can be
e. (negative) – (positive): Always
f. (negative) + (positive): Can be
9. This string has a total of 100 tokens arranged in a particular pattern. What is the value of the string?
The pattern shows: + + + – – + + + – – + + + – –...
The value is