Year 5: Missing digits and number puzzles
Part of the Year 5 column addition and subtraction chapter. Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Missing digits and number puzzles
In a missing digit puzzle some digits in a column calculation are hidden. Each box stands for one digit from 0 to 9. You can find the digits with logic, and there is only one right answer.
Addition puzzles. Start with the ones column on the right, like you always do. If the missing digit is in the answer, just add. If it is in one of the numbers, ask: “what do I add to the digit I know to get a number that ends in the answer digit?” Then think about the carry, because it matters in the next column.
| TTh | Th | H | T | O | |
| 4 | 5 | ? | 6 | 8 | |
| + | 2 | 7 | 5 | ? | 4 |
| 7 | 2 | 9 | 6 | ? |
Subtraction puzzles. Turn a subtraction into an addition. If 84,270 − 51,835 = 32,435, then 51,835 + 32,435 = 84,270. The inverse sum has the same digits, and you can solve it from the right.
| TTh | Th | H | T | O | |
| 8 | 4 | ? | 7 | 0 | |
| − | 5 | ? | 8 | 3 | 5 |
| 3 | 2 | 4 | 3 | ? |
Balance equations. The equals sign (=) means “is the same as”. It does not always have the answer on its right. In 63,500 + ? = 100,000 − 14,500, work out the side with no box first (85,500). Then find the missing number: 85,500 − 63,500 = 22,000.
Reasoning puzzles. Number-card puzzles ask you to think about the value of each column. A digit in a big column matters more than a digit in a small column. For a big total, put big digits in big columns. For a small difference, make the numbers as close as you can.
Key words. A digit is one of 0 to 9. An equation says that two amounts are equal. The inverse of an operation undoes it.
Key ideas
- Always start with the ones column and work to the left.
- Remember the carry. A missing digit must make the column total end in the right digit.
- Turn a subtraction puzzle into an addition with the inverse: top − bottom = answer means bottom + answer = top.
- The = sign means “is the same as”. Work out the side with no box first.
- For a big total put big digits in big columns. For a small difference make the numbers close.
- Check by doing the completed calculation.
| TTh | Th | H | T | O | |
| 4 | 5 | ? | 6 | 8 | |
| + | 2 | 7 | 5 | ? | 4 |
| 7 | 2 | 9 | 6 | ? |
Key words. A digit is one of 0 to 9. The ones column is the column on the right.
Method. Work from the ones column on the right. In each column use the digits you know, and the carry from the column before, to find the missing digit.
- Find the missing digit in the ones column of the answer. We know the digits we are adding, so we add them (and the carry).8 + 4 = 12. The missing digit is 2. Carry 1 to the tens column.
TTh Th H T O 4 5 ? 6 8 + 2 7 5 ? 4 7 2 9 6 2 - Find the missing digit in the tens column of the second number. The answer digit is 6, so the column total must end in 6. What do we add to the digits we know?? + 6 + 1 must end in 6.
Count on from 7 to a number that ends in 6: 7 + 9 = 16.
The missing digit is 9. Carry 1 to the hundreds column.TTh Th H T O 4 5 ? 6 8 + 2 7 5 9 4 7 2 9 6 2 - Find the missing digit in the hundreds column of the first number. The answer digit is 9, so the column total must end in 9. What do we add to the digits we know?? + 5 + 1 must end in 9.
Count on from 6 to a number that ends in 9: 6 + 3 = 9.
The missing digit is 3.TTh Th H T O 4 5 3 6 8 + 2 7 5 9 4 7 2 9 6 2 - Look at the thousands column. Nothing is hidden here, but we need to know the carry for the next column.5 + 7 = 12. Write 2. Carry 1 to the ten thousands column.
TTh Th H T O 4 5 3 6 8 + 2 7 5 9 4 7 2 9 6 2 - Look at the ten thousands column. Nothing is hidden here.4 + 2 + 1 = 7. Write 7.
TTh Th H T O 4 5 3 6 8 + 2 7 5 9 4 7 2 9 6 2 - Write the completed calculation. Put the missing digits back and read the whole calculation.
TTh Th H T O 4 5 3 6 8 + 2 7 5 9 4 7 2 9 6 2 1 1 1
45,368 + 27,594 = 72,962
answer, ones digit: 2
second number, tens digit: 9
first number, hundreds digit: 3
Check. Add again: 45,368 + 27,594 = 72,962. ✓
Watch out. Do not forget the carry. A missing digit must make the column total end in the right digit. Then the carry goes to the next column.
| HTh | TTh | Th | H | T | O | |
| ? | 5 | 8 | 7 | 6 | 4 | |
| + | 2 | 4 | ? | 6 | 8 | 9 |
| 6 | ? | 6 | 4 | ? | 3 |
Key words. A digit is one of 0 to 9. The ones column is the column on the right.
Method. Work from the ones column on the right. In each column use the digits you know, and the carry from the column before, to find the missing digit.
- Look at the ones column. Nothing is hidden here, but we need to know the carry for the next column.4 + 9 = 13. Write 3. Carry 1 to the tens column.
- Find the missing digit in the tens column of the answer. We know the digits we are adding, so we add them (and the carry).6 + 8 + 1 = 15. The missing digit is 5. Carry 1 to the hundreds column.
- Look at the hundreds column. Nothing is hidden here, but we need to know the carry for the next column.7 + 6 + 1 = 14. Write 4. Carry 1 to the thousands column.
- Find the missing digit in the thousands column of the second number. The answer digit is 6, so the column total must end in 6. What do we add to the digits we know?? + 8 + 1 must end in 6.
Count on from 9 to a number that ends in 6: 9 + 7 = 16.
The missing digit is 7. Carry 1 to the ten thousands column. - Find the missing digit in the ten thousands column of the answer. We know the digits we are adding, so we add them (and the carry).5 + 4 + 1 = 10. The missing digit is 0. Carry 1 to the hundred thousands column.
- Find the missing digit in the hundred thousands column of the first number. The answer digit is 6, so the column total must end in 6. What do we add to the digits we know?? + 2 + 1 must end in 6.
Count on from 3 to a number that ends in 6: 3 + 3 = 6.
The missing digit is 3. - Write the completed calculation. Put the missing digits back and read the whole calculation.
HTh TTh Th H T O 3 5 8 7 6 4 + 2 4 7 6 8 9 6 0 6 4 5 3 1 1 1 1 1
358,764 + 247,689 = 606,453
answer, tens digit: 5
second number, thousands digit: 7
answer, ten thousands digit: 0
first number, hundred thousands digit: 3
Check. Add again: 358,764 + 247,689 = 606,453. ✓
Watch out. Do not forget the carry. A missing digit must make the column total end in the right digit. Then the carry goes to the next column.
| TTh | Th | H | T | O | |
| 8 | 4 | ? | 7 | 0 | |
| − | 5 | ? | 8 | 3 | 5 |
| 3 | 2 | 4 | 3 | ? |
Key words. A digit is one of 0 to 9. The ones column is the column on the right.
Method. Use the inverse operation to turn the subtraction into an addition. Then work from the ones column on the right.
- Turn the subtraction into an addition. Subtraction is the inverse of addition: the answer plus the number taken away makes the number we started with. Now we can work from the right, like an addition.84,270 − 51,835 = 32,435 means 51,835 + 32,435 = 84,270
TTh Th H T O 5 ? 8 3 5 + 3 2 4 3 ? 8 4 ? 7 0 - Find the missing digit in the ones column of the answer. The answer digit is 0, so the column total must end in 0. What do we add to the digits we know?? + 5 must end in 0.
Count on from 5 to a number that ends in 0: 5 + 5 = 10.
The missing digit is 5. Carry 1 to the tens column.TTh Th H T O 5 ? 8 3 5 + 3 2 4 3 5 8 4 ? 7 0 - Look at the tens column. Nothing is hidden here.3 + 3 + 1 = 7. Write 7.
TTh Th H T O 5 ? 8 3 5 + 3 2 4 3 5 8 4 ? 7 0 - Find the missing digit in the hundreds column of the top number. We know the digits we are adding, so we add them (and the carry).8 + 4 = 12. The missing digit is 2. Carry 1 to the thousands column.
TTh Th H T O 5 ? 8 3 5 + 3 2 4 3 5 8 4 2 7 0 - Find the missing digit in the thousands column of the bottom number. The answer digit is 4, so the column total must end in 4. What do we add to the digits we know?? + 2 + 1 must end in 4.
Count on from 3 to a number that ends in 4: 3 + 1 = 4.
The missing digit is 1.TTh Th H T O 5 1 8 3 5 + 3 2 4 3 5 8 4 2 7 0 - Look at the ten thousands column. Nothing is hidden here.5 + 3 = 8. Write 8.
TTh Th H T O 5 1 8 3 5 + 3 2 4 3 5 8 4 2 7 0 - Write the completed calculation. Put the missing digits back and read the whole calculation.
TTh Th H T O 8 3 412 26 710 0− 5 1 8 3 5 3 2 4 3 5
84,270 − 51,835 = 32,435
answer, ones digit: 5
top number, hundreds digit: 2
bottom number, thousands digit: 1
Check. Subtract again with the completed numbers: 84,270 − 51,835 = 32,435. ✓
Watch out. Do not forget the carry. A missing digit must make the column total end in the right digit. Then the carry goes to the next column.
| HTh | TTh | Th | H | T | O | |
| 6 | 0 | 0 | 0 | ? | 0 | |
| − | 1 | ? | 4 | 5 | 6 | 8 |
| ? | 7 | 5 | ? | 3 | 2 |
Key words. A digit is one of 0 to 9. The ones column is the column on the right.
Method. Use the inverse operation to turn the subtraction into an addition. Then work from the ones column on the right.
- Turn the subtraction into an addition. Subtraction is the inverse of addition: the answer plus the number taken away makes the number we started with. Now we can work from the right, like an addition.600,000 − 124,568 = 475,432 means 124,568 + 475,432 = 600,000
HTh TTh Th H T O 1 ? 4 5 6 8 + ? 7 5 ? 3 2 6 0 0 0 ? 0 - Look at the ones column. Nothing is hidden here, but we need to know the carry for the next column.8 + 2 = 10. Write 0. Carry 1 to the tens column.
- Find the missing digit in the tens column of the top number. We know the digits we are adding, so we add them (and the carry).6 + 3 + 1 = 10. The missing digit is 0. Carry 1 to the hundreds column.
- Find the missing digit in the hundreds column of the answer. The answer digit is 0, so the column total must end in 0. What do we add to the digits we know?? + 5 + 1 must end in 0.
Count on from 6 to a number that ends in 0: 6 + 4 = 10.
The missing digit is 4. Carry 1 to the thousands column. - Look at the thousands column. Nothing is hidden here, but we need to know the carry for the next column.4 + 5 + 1 = 10. Write 0. Carry 1 to the ten thousands column.
- Find the missing digit in the ten thousands column of the bottom number. The answer digit is 0, so the column total must end in 0. What do we add to the digits we know?? + 7 + 1 must end in 0.
Count on from 8 to a number that ends in 0: 8 + 2 = 10.
The missing digit is 2. Carry 1 to the hundred thousands column. - Find the missing digit in the hundred thousands column of the answer. The answer digit is 6, so the column total must end in 6. What do we add to the digits we know?? + 1 + 1 must end in 6.
Count on from 2 to a number that ends in 6: 2 + 4 = 6.
The missing digit is 4. - Write the completed calculation. Put the missing digits back and read the whole calculation.
HTh TTh Th H T O 5 69 09 09 09 010 0− 1 2 4 5 6 8 4 7 5 4 3 2
600,000 − 124,568 = 475,432
top number, tens digit: 0
answer, hundreds digit: 4
bottom number, ten thousands digit: 2
answer, hundred thousands digit: 4
Check. Subtract again with the completed numbers: 600,000 − 124,568 = 475,432. ✓
Watch out. Do not forget the carry. A missing digit must make the column total end in the right digit. Then the carry goes to the next column.
Key words. An equation says that two amounts are equal. The left-hand side and right-hand side are the amounts on either side of the = sign.
Method. Both sides of the equals sign must have the same value. Work out the side with no box first.
- Look at the number sentence. The = sign means both sides are worth the same.63,500 + ? = 100,000 − 14,500
- Work out the right-hand side. It has no box, so we can work it out.100,000 − 14,500 = 85,500
HTh TTh Th H T O 0 19 09 010 00 0 − 1 4 5 0 0 8 5 5 0 0 - Find the missing number. To find a missing part, take the part we know away from the whole.85,500 − 63,500
85,500 − 63,500 = 22,000TTh Th H T O 8 5 5 0 0 − 6 3 5 0 0 2 2 0 0 0 - Check. Put the answer in the box and work out both sides.63,500 + 22,000 = 85,500 and the other side is 85,500. ✓
Check. Both sides give the same number. ✓
Watch out. The = sign does not mean “the answer is next”. It means “is the same as”, so a calculation can be on either side.
Method. Find out how much each 1 in the box changes the total. Then work out which digit is the first to make the sum true.
- Put 0 in the box and work out the sum. This gives the start of the list.6,500 + 2,480 = 8,980
- See what each extra 1 in the box does. The box is in the tens column, so each 1 more in the box makes the total 10 bigger.0 → 8,980, 1 → 8,990, 2 → 9,000, …
- Compare with 9,000. We need the sum to be more than 9,000. Check the digits next to the answer.box = 2: 6,520 + 2,480 = 9,000 (not more than 9,000)
box = 3: 6,530 + 2,480 = 9,010 > 9,000
box = 4: 6,540 + 2,480 = 9,020 > 9,000 - Write the answer. The smallest digit is 3.
Check. With 3 in the box the sum is 9,010, which is more than 9,000. ✓
Watch out. Read the question carefully: “more than” does not include the limit itself. If the sum is exactly equal to the limit, that digit does not work.
Method. To make the difference small, the two numbers must be close. Make the hundreds digits as close as possible, then make the top number as small as it can be and the bottom number as big as it can be.
- Choose the hundreds digits. The hundreds digits matter most. They should be next to each other in the list of digits.Digits in order: 1, 2, 3, 4, 5, 6
- Fill the other columns. The top number (the bigger one) gets the smallest digits that are left. The bottom number (the smaller one) gets the biggest digits that are left. Then the gap between them is as small as possible.hundreds digits 2 and 1: 234 − 165 = 69
hundreds digits 3 and 2: 314 − 265 = 49
hundreds digits 4 and 3: 412 − 365 = 47
hundreds digits 5 and 4: 512 − 463 = 49
hundreds digits 6 and 5: 612 − 543 = 69 - Find the smallest difference. Compare all the differences.The smallest is 412 − 365 = 47
Check. Check: 365 + 47 = 412. ✓ (Every other way of using the cards gives a bigger difference.)
Watch out. Use every card exactly once. Check that your two numbers use all six different digits.
Watch out
A box in a number sentence is not always at the end, and the answer is not always on the right of the = sign. In a puzzle, do not guess: use the ones column and the carry. And when you have an answer, always put the digits back and check the whole calculation.Exercise 2F Missing digits and puzzles
Non-calculator| TTh | Th | H | T | O | |
| 5 | ? | 4 | 1 | 2 | |
| + | 2 | 8 | 7 | ? | 6 |
| 8 | 2 | 1 | 5 | ? |
Key words. A digit is one of 0 to 9. The ones column is the column on the right.
Method. Work from the ones column on the right. In each column use the digits you know, and the carry from the column before, to find the missing digit.
- Find the missing digit in the ones column of the answer. We know the digits we are adding, so we add them (and the carry).2 + 6 = 8. The missing digit is 8.
- Find the missing digit in the tens column of the second number. The answer digit is 5, so the column total must end in 5. What do we add to the digits we know?? + 1 must end in 5.
Count on from 1 to a number that ends in 5: 1 + 4 = 5.
The missing digit is 4. - Look at the hundreds column. Nothing is hidden here, but we need to know the carry for the next column.4 + 7 = 11. Write 1. Carry 1 to the thousands column.
- Find the missing digit in the thousands column of the first number. The answer digit is 2, so the column total must end in 2. What do we add to the digits we know?? + 8 + 1 must end in 2.
Count on from 9 to a number that ends in 2: 9 + 3 = 12.
The missing digit is 3. Carry 1 to the ten thousands column. - Look at the ten thousands column. Nothing is hidden here.5 + 2 + 1 = 8. Write 8.
- Write the completed calculation. Put the missing digits back and read the whole calculation.
TTh Th H T O 5 3 4 1 2 + 2 8 7 4 6 8 2 1 5 8 1 1
53,412 + 28,746 = 82,158
answer, ones digit: 8
second number, tens digit: 4
first number, thousands digit: 3
Check. Add again: 53,412 + 28,746 = 82,158. ✓
Watch out. Do not forget the carry. A missing digit must make the column total end in the right digit. Then the carry goes to the next column.
| TTh | Th | H | T | O | |
| ? | 7 | 2 | 8 | 5 | |
| + | 3 | 6 | ? | 1 | 9 |
| 8 | 3 | 8 | ? | 4 |
Key words. A digit is one of 0 to 9. The ones column is the column on the right.
Method. Work from the ones column on the right. In each column use the digits you know, and the carry from the column before, to find the missing digit.
- Look at the ones column. Nothing is hidden here, but we need to know the carry for the next column.5 + 9 = 14. Write 4. Carry 1 to the tens column.
- Find the missing digit in the tens column of the answer. We know the digits we are adding, so we add them (and the carry).8 + 1 + 1 = 10. The missing digit is 0. Carry 1 to the hundreds column.
- Find the missing digit in the hundreds column of the second number. The answer digit is 8, so the column total must end in 8. What do we add to the digits we know?? + 2 + 1 must end in 8.
Count on from 3 to a number that ends in 8: 3 + 5 = 8.
The missing digit is 5. - Look at the thousands column. Nothing is hidden here, but we need to know the carry for the next column.7 + 6 = 13. Write 3. Carry 1 to the ten thousands column.
- Find the missing digit in the ten thousands column of the first number. The answer digit is 8, so the column total must end in 8. What do we add to the digits we know?? + 3 + 1 must end in 8.
Count on from 4 to a number that ends in 8: 4 + 4 = 8.
The missing digit is 4. - Write the completed calculation. Put the missing digits back and read the whole calculation.
TTh Th H T O 4 7 2 8 5 + 3 6 5 1 9 8 3 8 0 4 1 1 1
47,285 + 36,519 = 83,804
answer, tens digit: 0
second number, hundreds digit: 5
first number, ten thousands digit: 4
Check. Add again: 47,285 + 36,519 = 83,804. ✓
Watch out. Do not forget the carry. A missing digit must make the column total end in the right digit. Then the carry goes to the next column.
| TTh | Th | H | T | O | |
| 6 | 0 | 3 | 5 | ? | |
| + | 2 | ? | 8 | 7 | 4 |
| 9 | 0 | ? | 2 | 4 |
Key words. A digit is one of 0 to 9. The ones column is the column on the right.
Method. Work from the ones column on the right. In each column use the digits you know, and the carry from the column before, to find the missing digit.
- Find the missing digit in the ones column of the first number. The answer digit is 4, so the column total must end in 4. What do we add to the digits we know?? + 4 must end in 4.
Count on from 4 to a number that ends in 4: 4 + 0 = 4.
The missing digit is 0. - Look at the tens column. Nothing is hidden here, but we need to know the carry for the next column.5 + 7 = 12. Write 2. Carry 1 to the hundreds column.
- Find the missing digit in the hundreds column of the answer. We know the digits we are adding, so we add them (and the carry).3 + 8 + 1 = 12. The missing digit is 2. Carry 1 to the thousands column.
- Find the missing digit in the thousands column of the second number. The answer digit is 0, so the column total must end in 0. What do we add to the digits we know?? + 0 + 1 must end in 0.
Count on from 1 to a number that ends in 0: 1 + 9 = 10.
The missing digit is 9. Carry 1 to the ten thousands column. - Look at the ten thousands column. Nothing is hidden here.6 + 2 + 1 = 9. Write 9.
- Write the completed calculation. Put the missing digits back and read the whole calculation.
TTh Th H T O 6 0 3 5 0 + 2 9 8 7 4 9 0 2 2 4 1 1 1
60,350 + 29,874 = 90,224
first number, ones digit: 0
answer, hundreds digit: 2
second number, thousands digit: 9
Check. Add again: 60,350 + 29,874 = 90,224. ✓
Watch out. Do not forget the carry. A missing digit must make the column total end in the right digit. Then the carry goes to the next column.
| HTh | TTh | Th | H | T | O | |
| ? | 8 | 5 | 4 | 6 | 7 | |
| + | 2 | ? | 7 | 2 | 8 | 9 |
| 6 | 3 | 2 | ? | 5 | ? |
Key words. A digit is one of 0 to 9. The ones column is the column on the right.
Method. Work from the ones column on the right. In each column use the digits you know, and the carry from the column before, to find the missing digit.
- Find the missing digit in the ones column of the answer. We know the digits we are adding, so we add them (and the carry).7 + 9 = 16. The missing digit is 6. Carry 1 to the tens column.
- Look at the tens column. Nothing is hidden here, but we need to know the carry for the next column.6 + 8 + 1 = 15. Write 5. Carry 1 to the hundreds column.
- Find the missing digit in the hundreds column of the answer. We know the digits we are adding, so we add them (and the carry).4 + 2 + 1 = 7. The missing digit is 7.
- Look at the thousands column. Nothing is hidden here, but we need to know the carry for the next column.5 + 7 = 12. Write 2. Carry 1 to the ten thousands column.
- Find the missing digit in the ten thousands column of the second number. The answer digit is 3, so the column total must end in 3. What do we add to the digits we know?? + 8 + 1 must end in 3.
Count on from 9 to a number that ends in 3: 9 + 4 = 13.
The missing digit is 4. Carry 1 to the hundred thousands column. - Find the missing digit in the hundred thousands column of the first number. The answer digit is 6, so the column total must end in 6. What do we add to the digits we know?? + 2 + 1 must end in 6.
Count on from 3 to a number that ends in 6: 3 + 3 = 6.
The missing digit is 3. - Write the completed calculation. Put the missing digits back and read the whole calculation.
HTh TTh Th H T O 3 8 5 4 6 7 + 2 4 7 2 8 9 6 3 2 7 5 6 1 1 1 1
385,467 + 247,289 = 632,756
answer, ones digit: 6
answer, hundreds digit: 7
second number, ten thousands digit: 4
first number, hundred thousands digit: 3
Check. Add again: 385,467 + 247,289 = 632,756. ✓
Watch out. Do not forget the carry. A missing digit must make the column total end in the right digit. Then the carry goes to the next column.
| HTh | TTh | Th | H | T | O | |
| 5 | ? | 4 | 8 | 3 | 0 | |
| + | 3 | 9 | 6 | ? | 2 | 5 |
| ? | 0 | 1 | 5 | 5 | 5 |
Key words. A digit is one of 0 to 9. The ones column is the column on the right.
Method. Work from the ones column on the right. In each column use the digits you know, and the carry from the column before, to find the missing digit.
- Look at the ones column. Nothing is hidden here.0 + 5 = 5. Write 5.
- Look at the tens column. Nothing is hidden here.3 + 2 = 5. Write 5.
- Find the missing digit in the hundreds column of the second number. The answer digit is 5, so the column total must end in 5. What do we add to the digits we know?? + 8 must end in 5.
Count on from 8 to a number that ends in 5: 8 + 7 = 15.
The missing digit is 7. Carry 1 to the thousands column. - Look at the thousands column. Nothing is hidden here, but we need to know the carry for the next column.4 + 6 + 1 = 11. Write 1. Carry 1 to the ten thousands column.
- Find the missing digit in the ten thousands column of the first number. The answer digit is 0, so the column total must end in 0. What do we add to the digits we know?? + 9 + 1 must end in 0.
Count on from 10 to a number that ends in 0: 10 + 0 = 10.
The missing digit is 0. Carry 1 to the hundred thousands column. - Find the missing digit in the hundred thousands column of the answer. We know the digits we are adding, so we add them (and the carry).5 + 3 + 1 = 9. The missing digit is 9.
- Write the completed calculation. Put the missing digits back and read the whole calculation.
HTh TTh Th H T O 5 0 4 8 3 0 + 3 9 6 7 2 5 9 0 1 5 5 5 1 1 1
504,830 + 396,725 = 901,555
second number, hundreds digit: 7
first number, ten thousands digit: 0
answer, hundred thousands digit: 9
Check. Add again: 504,830 + 396,725 = 901,555. ✓
Watch out. Do not forget the carry. A missing digit must make the column total end in the right digit. Then the carry goes to the next column.
| HTh | TTh | Th | H | T | O | |
| 2 | 7 | ? | 9 | 1 | 5 | |
| + | ? | 6 | 3 | 0 | 8 | 9 |
| 7 | 4 | 2 | 0 | ? | 4 |
Key words. A digit is one of 0 to 9. The ones column is the column on the right.
Method. Work from the ones column on the right. In each column use the digits you know, and the carry from the column before, to find the missing digit.
- Look at the ones column. Nothing is hidden here, but we need to know the carry for the next column.5 + 9 = 14. Write 4. Carry 1 to the tens column.
- Find the missing digit in the tens column of the answer. We know the digits we are adding, so we add them (and the carry).1 + 8 + 1 = 10. The missing digit is 0. Carry 1 to the hundreds column.
- Look at the hundreds column. Nothing is hidden here, but we need to know the carry for the next column.9 + 0 + 1 = 10. Write 0. Carry 1 to the thousands column.
- Find the missing digit in the thousands column of the first number. The answer digit is 2, so the column total must end in 2. What do we add to the digits we know?? + 3 + 1 must end in 2.
Count on from 4 to a number that ends in 2: 4 + 8 = 12.
The missing digit is 8. Carry 1 to the ten thousands column. - Look at the ten thousands column. Nothing is hidden here, but we need to know the carry for the next column.7 + 6 + 1 = 14. Write 4. Carry 1 to the hundred thousands column.
- Find the missing digit in the hundred thousands column of the second number. The answer digit is 7, so the column total must end in 7. What do we add to the digits we know?? + 2 + 1 must end in 7.
Count on from 3 to a number that ends in 7: 3 + 4 = 7.
The missing digit is 4. - Write the completed calculation. Put the missing digits back and read the whole calculation.
HTh TTh Th H T O 2 7 8 9 1 5 + 4 6 3 0 8 9 7 4 2 0 0 4 1 1 1 1 1
278,915 + 463,089 = 742,004
answer, tens digit: 0
first number, thousands digit: 8
second number, hundred thousands digit: 4
Check. Add again: 278,915 + 463,089 = 742,004. ✓
Watch out. Do not forget the carry. A missing digit must make the column total end in the right digit. Then the carry goes to the next column.
| TTh | Th | H | T | O | |
| 9 | ? | 5 | 0 | 0 | |
| − | ? | 7 | 2 | 6 | 8 |
| 4 | 3 | 2 | ? | 2 |
Key words. A digit is one of 0 to 9. The ones column is the column on the right.
Method. Use the inverse operation to turn the subtraction into an addition. Then work from the ones column on the right.
- Turn the subtraction into an addition. Subtraction is the inverse of addition: the answer plus the number taken away makes the number we started with. Now we can work from the right, like an addition.90,500 − 47,268 = 43,232 means 47,268 + 43,232 = 90,500
TTh Th H T O ? 7 2 6 8 + 4 3 2 ? 2 9 ? 5 0 0 - Look at the ones column. Nothing is hidden here, but we need to know the carry for the next column.8 + 2 = 10. Write 0. Carry 1 to the tens column.
- Find the missing digit in the tens column of the answer. The answer digit is 0, so the column total must end in 0. What do we add to the digits we know?? + 6 + 1 must end in 0.
Count on from 7 to a number that ends in 0: 7 + 3 = 10.
The missing digit is 3. Carry 1 to the hundreds column. - Look at the hundreds column. Nothing is hidden here.2 + 2 + 1 = 5. Write 5.
- Find the missing digit in the thousands column of the top number. We know the digits we are adding, so we add them (and the carry).7 + 3 = 10. The missing digit is 0. Carry 1 to the ten thousands column.
- Find the missing digit in the ten thousands column of the bottom number. The answer digit is 9, so the column total must end in 9. What do we add to the digits we know?? + 4 + 1 must end in 9.
Count on from 5 to a number that ends in 9: 5 + 4 = 9.
The missing digit is 4. - Write the completed calculation. Put the missing digits back and read the whole calculation.
TTh Th H T O 8 910 04 59 010 0− 4 7 2 6 8 4 3 2 3 2
90,500 − 47,268 = 43,232
answer, tens digit: 3
top number, thousands digit: 0
bottom number, ten thousands digit: 4
Check. Subtract again with the completed numbers: 90,500 − 47,268 = 43,232. ✓
Watch out. Do not forget the carry. A missing digit must make the column total end in the right digit. Then the carry goes to the next column.
| TTh | Th | H | T | O | |
| 7 | 3 | ? | 0 | 5 | |
| − | 2 | 8 | 6 | 7 | ? |
| 4 | ? | 7 | 2 | 8 |
Key words. A digit is one of 0 to 9. The ones column is the column on the right.
Method. Use the inverse operation to turn the subtraction into an addition. Then work from the ones column on the right.
- Turn the subtraction into an addition. Subtraction is the inverse of addition: the answer plus the number taken away makes the number we started with. Now we can work from the right, like an addition.73,405 − 28,677 = 44,728 means 28,677 + 44,728 = 73,405
TTh Th H T O 2 8 6 7 ? + 4 ? 7 2 8 7 3 ? 0 5 - Find the missing digit in the ones column of the bottom number. The answer digit is 5, so the column total must end in 5. What do we add to the digits we know?? + 8 must end in 5.
Count on from 8 to a number that ends in 5: 8 + 7 = 15.
The missing digit is 7. Carry 1 to the tens column. - Look at the tens column. Nothing is hidden here, but we need to know the carry for the next column.7 + 2 + 1 = 10. Write 0. Carry 1 to the hundreds column.
- Find the missing digit in the hundreds column of the top number. We know the digits we are adding, so we add them (and the carry).6 + 7 + 1 = 14. The missing digit is 4. Carry 1 to the thousands column.
- Find the missing digit in the thousands column of the answer. The answer digit is 3, so the column total must end in 3. What do we add to the digits we know?? + 8 + 1 must end in 3.
Count on from 9 to a number that ends in 3: 9 + 4 = 13.
The missing digit is 4. Carry 1 to the ten thousands column. - Look at the ten thousands column. Nothing is hidden here.2 + 4 + 1 = 7. Write 7.
- Write the completed calculation. Put the missing digits back and read the whole calculation.
TTh Th H T O 6 712 313 49 015 5− 2 8 6 7 7 4 4 7 2 8
73,405 − 28,677 = 44,728
bottom number, ones digit: 7
top number, hundreds digit: 4
answer, thousands digit: 4
Check. Subtract again with the completed numbers: 73,405 − 28,677 = 44,728. ✓
Watch out. Do not forget the carry. A missing digit must make the column total end in the right digit. Then the carry goes to the next column.
| HTh | TTh | Th | H | T | O | |
| ? | 1 | 7 | 2 | 4 | 0 | |
| − | 3 | 5 | ? | 1 | 8 | 6 |
| 4 | ? | 8 | 0 | 5 | ? |
Key words. A digit is one of 0 to 9. The ones column is the column on the right.
Method. Use the inverse operation to turn the subtraction into an addition. Then work from the ones column on the right.
- Turn the subtraction into an addition. Subtraction is the inverse of addition: the answer plus the number taken away makes the number we started with. Now we can work from the right, like an addition.817,240 − 359,186 = 458,054 means 359,186 + 458,054 = 817,240
HTh TTh Th H T O 3 5 ? 1 8 6 + 4 ? 8 0 5 ? ? 1 7 2 4 0 - Find the missing digit in the ones column of the answer. The answer digit is 0, so the column total must end in 0. What do we add to the digits we know?? + 6 must end in 0.
Count on from 6 to a number that ends in 0: 6 + 4 = 10.
The missing digit is 4. Carry 1 to the tens column. - Look at the tens column. Nothing is hidden here, but we need to know the carry for the next column.8 + 5 + 1 = 14. Write 4. Carry 1 to the hundreds column.
- Look at the hundreds column. Nothing is hidden here.1 + 0 + 1 = 2. Write 2.
- Find the missing digit in the thousands column of the bottom number. The answer digit is 7, so the column total must end in 7. What do we add to the digits we know?? + 8 must end in 7.
Count on from 8 to a number that ends in 7: 8 + 9 = 17.
The missing digit is 9. Carry 1 to the ten thousands column. - Find the missing digit in the ten thousands column of the answer. The answer digit is 1, so the column total must end in 1. What do we add to the digits we know?? + 5 + 1 must end in 1.
Count on from 6 to a number that ends in 1: 6 + 5 = 11.
The missing digit is 5. Carry 1 to the hundred thousands column. - Find the missing digit in the hundred thousands column of the top number. We know the digits we are adding, so we add them (and the carry).3 + 4 + 1 = 8. The missing digit is 8.
- Write the completed calculation. Put the missing digits back and read the whole calculation.
HTh TTh Th H T O 7 810 117 71 213 410 0− 3 5 9 1 8 6 4 5 8 0 5 4
817,240 − 359,186 = 458,054
answer, ones digit: 4
bottom number, thousands digit: 9
answer, ten thousands digit: 5
top number, hundred thousands digit: 8
Check. Subtract again with the completed numbers: 817,240 − 359,186 = 458,054. ✓
Watch out. Do not forget the carry. A missing digit must make the column total end in the right digit. Then the carry goes to the next column.
| HTh | TTh | Th | H | T | O | |
| 5 | 0 | 0 | 0 | 0 | ? | |
| − | 2 | 1 | ? | 4 | 7 | 5 |
| ? | 8 | 6 | ? | 2 | 5 |
Key words. A digit is one of 0 to 9. The ones column is the column on the right.
Method. Use the inverse operation to turn the subtraction into an addition. Then work from the ones column on the right.
- Turn the subtraction into an addition. Subtraction is the inverse of addition: the answer plus the number taken away makes the number we started with. Now we can work from the right, like an addition.500,000 − 213,475 = 286,525 means 213,475 + 286,525 = 500,000
HTh TTh Th H T O 2 1 ? 4 7 5 + ? 8 6 ? 2 5 5 0 0 0 0 ? - Find the missing digit in the ones column of the top number. We know the digits we are adding, so we add them (and the carry).5 + 5 = 10. The missing digit is 0. Carry 1 to the tens column.
- Look at the tens column. Nothing is hidden here, but we need to know the carry for the next column.7 + 2 + 1 = 10. Write 0. Carry 1 to the hundreds column.
- Find the missing digit in the hundreds column of the answer. The answer digit is 0, so the column total must end in 0. What do we add to the digits we know?? + 4 + 1 must end in 0.
Count on from 5 to a number that ends in 0: 5 + 5 = 10.
The missing digit is 5. Carry 1 to the thousands column. - Find the missing digit in the thousands column of the bottom number. The answer digit is 0, so the column total must end in 0. What do we add to the digits we know?? + 6 + 1 must end in 0.
Count on from 7 to a number that ends in 0: 7 + 3 = 10.
The missing digit is 3. Carry 1 to the ten thousands column. - Look at the ten thousands column. Nothing is hidden here, but we need to know the carry for the next column.1 + 8 + 1 = 10. Write 0. Carry 1 to the hundred thousands column.
- Find the missing digit in the hundred thousands column of the answer. The answer digit is 5, so the column total must end in 5. What do we add to the digits we know?? + 2 + 1 must end in 5.
Count on from 3 to a number that ends in 5: 3 + 2 = 5.
The missing digit is 2. - Write the completed calculation. Put the missing digits back and read the whole calculation.
HTh TTh Th H T O 4 59 09 09 09 010 0− 2 1 3 4 7 5 2 8 6 5 2 5
500,000 − 213,475 = 286,525
top number, ones digit: 0
answer, hundreds digit: 5
bottom number, thousands digit: 3
answer, hundred thousands digit: 2
Check. Subtract again with the completed numbers: 500,000 − 213,475 = 286,525. ✓
Watch out. Do not forget the carry. A missing digit must make the column total end in the right digit. Then the carry goes to the next column.
Key words. An equation says that two amounts are equal. The left-hand side and right-hand side are the amounts on either side of the = sign.
Method. Both sides of the equals sign must have the same value. Work out the side with no box first.
- Look at the number sentence. The = sign means both sides are worth the same.63,500 + ? = 100,000 − 14,500
- Work out the right-hand side. It has no box, so we can work it out.100,000 − 14,500 = 85,500
HTh TTh Th H T O 0 19 09 010 00 0 − 1 4 5 0 0 8 5 5 0 0 - Find the missing number. To find a missing part, take the part we know away from the whole.85,500 − 63,500
85,500 − 63,500 = 22,000TTh Th H T O 8 5 5 0 0 − 6 3 5 0 0 2 2 0 0 0 - Check. Put the answer in the box and work out both sides.63,500 + 22,000 = 85,500 and the other side is 85,500. ✓
Check. Both sides give the same number. ✓
Watch out. The = sign does not mean “the answer is next”. It means “is the same as”, so a calculation can be on either side.
Key words. An equation says that two amounts are equal. The left-hand side and right-hand side are the amounts on either side of the = sign.
Method. Both sides of the equals sign must have the same value. Work out the side with no box first.
- Look at the number sentence. The = sign means both sides are worth the same.250,000 − ? = 180,000 + 35,000
- Work out the right-hand side. It has no box, so we can work it out.180,000 + 35,000 = 215,000
HTh TTh Th H T O 1 8 0 0 0 0 + 3 5 0 0 0 2 1 5 0 0 0 1 - Find the missing number. The first number is the whole. The missing number is one part, and the right-hand side is the other part.250,000 − 215,000
250,000 − 215,000 = 35,000HTh TTh Th H T O 2 4 510 00 0 0 − 2 1 5 0 0 0 3 5 0 0 0 - Check. Put the answer in the box and work out both sides.250,000 − 35,000 = 215,000 and the other side is 215,000. ✓
Check. Both sides give the same number. ✓
Watch out. The = sign does not mean “the answer is next”. It means “is the same as”, so a calculation can be on either side.
Key words. An equation says that two amounts are equal. The left-hand side and right-hand side are the amounts on either side of the = sign.
Method. Both sides of the equals sign must have the same value. Work out the side with no box first.
- Look at the number sentence. The = sign means both sides are worth the same.? − 48,750 = 90,000 − 25,000
- Work out the right-hand side. It has no box, so we can work it out.90,000 − 25,000 = 65,000
TTh Th H T O 8 910 00 0 0 − 2 5 0 0 0 6 5 0 0 0 - Find the missing number. The missing number is the whole. The two parts are the right-hand side and the number taken away. Add them.65,000 + 48,750
65,000 + 48,750 = 113,750HTh TTh Th H T O 6 5 0 0 0 + 4 8 7 5 0 1 1 3 7 5 0 1 1 - Check. Put the answer in the box and work out both sides.113,750 − 48,750 = 65,000 and the other side is 65,000. ✓
Check. Both sides give the same number. ✓
Watch out. The = sign does not mean “the answer is next”. It means “is the same as”, so a calculation can be on either side.
Key words. An equation says that two amounts are equal. The left-hand side and right-hand side are the amounts on either side of the = sign.
Method. Both sides of the equals sign must have the same value. Work out the side with no box first.
- Look at the number sentence. The = sign means both sides are worth the same.7,450 + ? = 3,000 + 9,999
- Work out the right-hand side. It has no box, so we can work it out.3,000 + 9,999 = 12,999
TTh Th H T O 3 0 0 0 + 9 9 9 9 1 2 9 9 9 1 - Find the missing number. To find a missing part, take the part we know away from the whole.12,999 − 7,450
12,999 − 7,450 = 5,549TTh Th H T O 0 112 29 9 9 − 7 4 5 0 5 5 4 9 - Check. Put the answer in the box and work out both sides.7,450 + 5,549 = 12,999 and the other side is 12,999. ✓
Check. Both sides give the same number. ✓
Watch out. The = sign does not mean “the answer is next”. It means “is the same as”, so a calculation can be on either side.
Method. Find out how much each 1 in the box changes the total. Then work out which digit is the first to make the sum true.
- Put 0 in the box and work out the sum. This gives the start of the list.6,500 + 2,480 = 8,980
- See what each extra 1 in the box does. The box is in the tens column, so each 1 more in the box makes the total 10 bigger.0 → 8,980, 1 → 8,990, 2 → 9,000, …
- Compare with 9,000. We need the sum to be more than 9,000. Check the digits next to the answer.box = 2: 6,520 + 2,480 = 9,000 (not more than 9,000)
box = 3: 6,530 + 2,480 = 9,010 > 9,000
box = 4: 6,540 + 2,480 = 9,020 > 9,000 - Write the answer. The smallest digit is 3.
Check. With 3 in the box the sum is 9,010, which is more than 9,000. ✓
Watch out. Read the question carefully: “more than” does not include the limit itself. If the sum is exactly equal to the limit, that digit does not work.
Method. Find out how much each 1 in the box changes the total. Then work out which digit is the first to make the sum true.
- Put 0 in the box and work out the sum. This gives the start of the list.4,075 + 38,600 = 42,675
- See what each extra 1 in the box does. The box is in the hundreds column, so each 1 more in the box makes the total 100 bigger.0 → 42,675, 1 → 42,775, 2 → 42,875, …
- Compare with 43,000. We need the sum to be less than 43,000. Check the digits next to the answer.box = 2: 4,275 + 38,600 = 42,875 < 43,000
box = 3: 4,375 + 38,600 = 42,975 < 43,000
box = 4: 4,475 + 38,600 = 43,075 (not less than 43,000) - Write the answer. The largest digit is 3.
Check. With 3 in the box the sum is 42,975, which is less than 43,000. ✓
Watch out. Read the question carefully: “more than” does not include the limit itself. If the sum is exactly equal to the limit, that digit does not work.
Method. Find out how much each 1 in the box changes the total. Then work out which digit is the first to make the sum true.
- Put 0 in the box and work out the sum. This gives the start of the list.2,705 + 56,090 = 58,795
- See what each extra 1 in the box does. The box is in the tens column, so each 1 more in the box makes the total 10 bigger.0 → 58,795, 1 → 58,805, 2 → 58,815, …
- Compare with 58,800. We need the sum to be more than 58,800. Check the digits next to the answer.box = 0: 2,705 + 56,090 = 58,795 (not more than 58,800)
box = 1: 2,715 + 56,090 = 58,805 > 58,800
box = 2: 2,725 + 56,090 = 58,815 > 58,800 - Write the answer. The smallest digit is 1.
Check. With 1 in the box the sum is 58,805, which is more than 58,800. ✓
Watch out. Read the question carefully: “more than” does not include the limit itself. If the sum is exactly equal to the limit, that digit does not work.
Key words. To work backwards is to start from the answer and use the inverse of each step in reverse order.
Method. Work backwards. Use the inverse (opposite) of each step, starting from the last one.
- Write the steps in order. Dev did: start → add 18,450 → subtract 7,300 → 50,000.? → + 18,450 → − 7,300 → 50,000
- Undo the last step. The last step was − 7,300. The inverse is + 7,300.50,000 + 7,300 = 57,300
TTh Th H T O 5 0 0 0 0 + 7 3 0 0 5 7 3 0 0 - Undo the first step. The first step was + 18,450. The inverse is − 18,450.57,300 − 18,450 = 38,850
TTh Th H T O 4 516 712 310 00 − 1 8 4 5 0 3 8 8 5 0 - Write the number. 38,850
Check. Go forward: 38,850 + 18,450 = 57,300 and 57,300 − 7,300 = 50,000. ✓
Watch out. Work backwards, not forwards, and use the opposite operation each time.
| TTh | Th | H | T | O | |
| 6 | 3 | 4 | ? | 8 | |
| + | 2 | ? | 3 | 5 | 9 |
| 9 | 0 | 7 | 8 | ? |
Method. Check Hana’s digit by putting it back in the sum, then find out what she did wrong.
- Try Hana’s digit. Put 3 in the box and add the whole sum.63,438 + 27,359 = 90,797
The answer should be 90,787, so 3 is wrong. - Look at the ones column. There is a carry into the tens column.8 + 9 = 17, so write 7 and carry 1.
- Find the mistake. The tens column is ? + 5 + 1 (the carry), and it must end in 8. Hana did 8 − 5 and forgot the carry.? + 5 + 1 must end in 8. 6 + 2 = 8, so ? = 2.
- Write the correct digit. The digit is 2. 63,428 + 27,359 = 90,787
Check. 63,428 + 27,359 = 90,787. ✓
Watch out. In a puzzle, always add the carry from the column on the right before you find the missing digit.
Method. Check just the ones digit. The ones digit of the answer only depends on the ones digits of the two numbers. If it is wrong, the answer cannot be right.
- Look at the ones digits. The ones digits are 8 and 9.4,358 + 2,719: ones digits 8 and 9
- Work out the ones digit of the answer. We only need the ones digit. If the answer is 10 or more, the ones digit is the last digit.8 + 9 = 17, so the ones digit is 7
- Compare with the claimed answer. The claimed answer 7,075 has ones digit 5.5 is not the same as 7. So it cannot be right.
- Write the answer. Cannot be right (the ones digit should be 7)
Check. The correct answer is 7,077. 7,075 is wrong. ✓
Watch out. A matching ones digit does not prove an answer is right. It only helps you spot some wrong answers quickly.
Method. Check just the ones digit. The ones digit of the answer only depends on the ones digits of the two numbers. If it is wrong, the answer cannot be right.
- Look at the ones digits. The ones digits are 3 and 8.6,003 − 2,848: ones digits 3 and 8
- Work out the ones digit of the answer. We only need the ones digit. If the top digit is smaller we exchange first.13 − 8 = 5, so the ones digit is 5
- Compare with the claimed answer. The claimed answer 3,155 has ones digit 5.5 is the same as 5. It could be right (we would need to check the other digits too).
- Write the answer. Could be right
Check. The correct answer is 3,155. The claim 3,155 is right.
Watch out. A matching ones digit does not prove an answer is right. It only helps you spot some wrong answers quickly.
Method. Check just the ones digit. The ones digit of the answer only depends on the ones digits of the two numbers. If it is wrong, the answer cannot be right.
- Look at the ones digits. The ones digits are 6 and 9.52,346 + 38,179: ones digits 6 and 9
- Work out the ones digit of the answer. We only need the ones digit. If the answer is 10 or more, the ones digit is the last digit.6 + 9 = 15, so the ones digit is 5
- Compare with the claimed answer. The claimed answer 90,535 has ones digit 5.5 is the same as 5. It could be right (we would need to check the other digits too).
- Write the answer. Could be right
Check. The correct answer is 90,525. The claim 90,535 is wrong, even though the ones digit matches.
Watch out. A matching ones digit does not prove an answer is right. It only helps you spot some wrong answers quickly.
Method. Check just the ones digit. The ones digit of the answer only depends on the ones digits of the two numbers. If it is wrong, the answer cannot be right.
- Look at the ones digits. The ones digits are 0 and 7.80,420 − 34,867: ones digits 0 and 7
- Work out the ones digit of the answer. We only need the ones digit. If the top digit is smaller we exchange first.10 − 7 = 3, so the ones digit is 3
- Compare with the claimed answer. The claimed answer 45,556 has ones digit 6.6 is not the same as 3. So it cannot be right.
- Write the answer. Cannot be right (the ones digit should be 3)
Check. The correct answer is 45,553. 45,556 is wrong. ✓
Watch out. A matching ones digit does not prove an answer is right. It only helps you spot some wrong answers quickly.
Method. Try some examples. Look for one where both numbers end in 0 and one where they do not.
- Try 40 + 50. Both numbers end in 0.40 + 50 = 90. The answer ends in 0, and both numbers end in 0. It works.
- Try 17 + 23. The ones digits are 7 and 3. 7 + 3 = 10 is a number bond that ends in 0.17 + 23 = 40. The answer ends in 0, but the numbers end in 7 and 3. It does not work.
- Decide. We found one example of each.Sometimes
Check. Try 6,540 + 7,260 = 13,800 (both end in 0) and 6,547 + 7,263 = 13,810 (they do not). ✓
Watch out. “Sometimes” needs one example that works and one that does not.
Method. To make the difference as big as possible, make the top number as big as you can and the bottom number as small as you can.
- Make the biggest number. Put the biggest digits in the biggest columns.Biggest three digits: 9, 7, 6 → 976
- Make the smallest number. Use the three smallest digits left, with the smallest in the biggest column.Smallest three digits: 1, 3, 4 → 134
- Subtract. Biggest minus smallest.
H T O 9 7 6 − 1 3 4 8 4 2
976 − 134 = 842
Check. 134 + 842 = 976. ✓
Watch out. Every card must be used once. The bigger number goes on top.
Method. To make the difference small, the two numbers must be close. Make the hundreds digits as close as possible, then make the top number as small as it can be and the bottom number as big as it can be.
- Choose the hundreds digits. The hundreds digits matter most. They should be next to each other in the list of digits.Digits in order: 2, 3, 5, 6, 8, 9
- Fill the other columns. The top number (the bigger one) gets the smallest digits that are left. The bottom number (the smaller one) gets the biggest digits that are left. Then the gap between them is as small as possible.hundreds digits 3 and 2: 356 − 298 = 58
hundreds digits 5 and 3: 526 − 398 = 128
hundreds digits 6 and 5: 623 − 598 = 25
hundreds digits 8 and 6: 823 − 695 = 128
hundreds digits 9 and 8: 923 − 865 = 58 - Find the smallest difference. Compare all the differences.The smallest is 623 − 598 = 25
Check. Check: 598 + 25 = 623. ✓ (Every other way of using the cards gives a bigger difference.)
Watch out. Use every card exactly once. Check that your two numbers use all six different digits.