Large Numbers Around Us · Lesson 2 of 9
Land of Tens: Building Numbers with Place Value
“Use imaginary calculators to discover number representations and efficient regrouping.”
• Find which numbers an addition-only calculator can reach from zero. • Represent the same number with different combinations of place-value buttons. • Explain why regrouping reduces button presses when all needed buttons exist. • Distinguish an exact press-count puzzle from a minimum-press problem. • Solve restricted-button challenges without applying an unsuitable shortcut.
One button, many equal steps
Imagine a calculator whose display starts at zero and whose only button adds a fixed number. Pressing it repeatedly is repeated addition: each press contributes another equal-sized group. This connects button presses to multiplication and division.
Thoughtful Thousands has only +1,000. Three presses display 3,000 and 53 presses display 53,000. To reverse the question, divide the target by 1,000. A target is reachable only when this gives a whole number of presses; there is no subtraction button and no fraction of a press.
| Calculator | Targets → required presses |
|---|---|
| +1,000 | 3,000 → 3; 10,000 → 10; 53,000 → 53; 90,000 → 90; 1,00,000 → 100; 1,53,000 → 153 |
| +10 | 500 → 50; 780 → 78; 1,000 → 100; 3,700 → 370; 10,000 → 1,000; 1,00,000 → 10,000; 4,350 → 435 |
| +100 | 400 → 4; 3,700 → 37; 10,000 → 100; 53,000 → 530; 90,000 → 900; 97,600 → 976; 1,00,000 → 1,000; 58,200 → 582 |
A multiple of a number is obtained by multiplying it by a whole number. From zero, a single addition button reaches exactly the non-negative multiples of its button value.
Problem
Can Handy Hundreds, with only +100, reach a number that neither +10 nor +1,000 can reach?
- 1.A number made by +100 presses has the form 100 × a whole number.
- 2.Each +100 press can be replaced by ten +10 presses. Therefore +10 can reach every number that +100 reaches.
- 3.For example, 3,700 needs 37 hundreds or 370 tens, but it is not a whole number of thousands.
- 4.Handy Hundreds can beat +1,000 on reachability, but cannot reach anything unavailable to both other calculators.
Many representations of one number
Creative Chitti has buttons +1, +10, +100, +1,000, +10,000, +1,00,000, and +10,00,000. There are now many ways to reach a target. We can replace one large group with ten smaller groups without changing the displayed number.
The number 321 can be made with 32 tens and 1 one: 320 + 1. It can also be made with 2 hundreds, 12 tens, and 1 one: 200 + 120 + 1. These group counts need not be single digits; an expression can describe repeated presses even when it is not the usual place-value expansion.
Problem
Check two non-standard representations of 5,072.
- 1.50 × 100 + 7 × 10 + 2 × 1 = 5,000 + 70 + 2 = 5,072.
- 2.3 × 1,000 + 20 × 100 + 72 × 1 = 3,000 + 2,000 + 72 = 5,072.
- 3.The first representation uses 50 + 7 + 2 = 59 presses; the second uses 3 + 20 + 72 = 95.
- 4.Equal displayed values can have different press counts. Always add the coefficients to count presses.
| Target | Place-value representation | Another valid representation |
|---|---|---|
| 8,300 | 8 × 1,000 + 3 × 100 | 83 × 100 |
| 40,629 | 4 × 10,000 + 6 × 100 + 2 × 10 + 9 | 40 × 1,000 + 62 × 10 + 9 |
| 56,354 | 5 × 10,000 + 6 × 1,000 + 3 × 100 + 5 × 10 + 4 | 56 × 1,000 + 35 × 10 + 4 |
| 66,666 | 6 × 10,000 + 6 × 1,000 + 6 × 100 + 6 × 10 + 6 | 666 × 100 + 6 × 10 + 6 |
| 3,67,813 | 3 × 1,00,000 + 6 × 10,000 + 7 × 1,000 + 8 × 100 + 1 × 10 + 3 | 36 × 10,000 + 78 × 100 + 13 |
Check the units of every group in the last row. Seventy-eight hundreds is 7,800; seventy-eight thousands is 78,000 and would produce a different total. A correct expression must pass two checks: its value equals the target and its coefficients describe the claimed number of presses.
Sippy finds the fewest presses
Sippy wants to reach the target efficiently. Suppose you have used at least ten presses of a small button and the next larger button exists. Ten +10 presses can become one +100 press, preserving the value while reducing the count by nine.
Repeat these exchanges until fewer than ten presses remain at every place. With all required place-value buttons available, the resulting counts are exactly the digits of the number. No further exchange is possible, and any ungrouped alternative would contain more presses. This explains why adding the digits gives the minimum in this particular calculator.
Problem
A student makes 5,072 using 5 thousands, 6 tens, and 12 ones. Find a shorter route.
- 1.The value is 5,000 + 60 + 12 = 5,072; the count is 5 + 6 + 12 = 23 presses.
- 2.Exchange ten of the twelve ones for one ten. The counts become 5 thousands, 7 tens, and 2 ones.
- 3.This uses 5 + 7 + 2 = 14 presses, nine fewer.
- 4.The usual expansion 5,000 + 70 + 2 therefore gives the minimum of 14 with Chitti’s buttons.
| Target | Minimum presses |
|---|---|
| 8,300 | 8 + 3 = 11 |
| 40,629 | 4 + 6 + 2 + 9 = 21 |
| 56,354 | 5 + 6 + 3 + 5 + 4 = 23 |
| 66,666 | 6 + 6 + 6 + 6 + 6 = 30 |
| 3,67,813 | 3 + 6 + 7 + 8 + 1 + 3 = 28 |
For 997, the usual representation uses 9 hundreds, 9 tens, and 7 ones: 25 presses. Replacing a hundred with ten tens gives 8 hundreds, 19 tens, and 7 ones: 34 presses. Ten presses of +10,00,000 give 1,00,00,000, a new grouping called one crore; a calculator without a crore button must keep those ten presses.
Exactly thirty is a different challenge
A minimum problem allows us to use fewer presses whenever possible. An exact-count problem requires the stated number of presses even if another route is shorter. To make a three-digit number with exactly 30 of Chitti’s presses, only +1, +10, and +100 can be used, since a larger button already reaches at least 1,000.
Problem
Make the largest and smallest three-digit numbers using exactly 30 presses.
- 1.Start by imagining all 30 presses as +1, giving 30. Replacing a +1 by +10 adds 9 to the result; replacing it by +100 adds 99, also a multiple of 9.
- 2.Every possible result therefore differs from 30 by a multiple of 9. Among three-digit numbers, the smallest such candidate is 102 and the largest is 993.
- 3.102 is achievable with 8 tens and 22 ones: 80 + 22 = 102 using 30 presses.
- 4.993 is achievable with 8 hundreds, 19 tens, and 3 ones: 800 + 190 + 3 = 993 using 30 presses.
- 5.Since each candidate is attainable and no more extreme three-digit number satisfies the necessary remainder condition, these are the required extremes.
When buttons are missing
Now keep only +10,000 and +100. Every reachable number must be a multiple of 100, and regrouping is possible only when 100 hundred-presses can become one ten-thousand press. The digit-sum shortcut does not apply because the intermediate +1,000 and smaller buttons are absent.
Problem
Find a representation of 92,100 using only +10,000 and +100.
- 1.Use nine +10,000 presses to make 90,000. The remaining value is 2,100.
- 2.2,100 ÷ 100 = 21, so use 21 +100 presses.
- 3.9 × 10,000 + 21 × 100 = 92,100, using 30 presses.
- 4.A tenth ten-thousand press would overshoot. Using fewer ten-thousand presses requires 100 extra hundred-presses for each one removed, so 30 is minimal.
| Target | Restricted-button expression | Press count |
|---|---|---|
| 20,800 | 2 × 10,000 + 8 × 100 | 10 |
| 92,100 | 9 × 10,000 + 21 × 100 | 30 |
| 1,20,500 | 12 × 10,000 + 5 × 100 | 17 |
| 65,30,000 | 653 × 10,000 | 653 |
| 70,25,700 | 702 × 10,000 + 57 × 100 | 759 |
Digit sum gives the minimum only when all needed decimal place-value buttons exist. With only +10,000 and +100, 20,800 needs 10 presses, although its digit sum is 10 too; this accidental agreement does not make the rule valid for every target. For 92,100 the digit sum is 12 but the required minimum is 30.
Quiz
Which target cannot be reached from zero with only +100?
How many +1,000 presses reach one lakh?
Which is a correct representation of 3,67,813?
With all needed decimal buttons, the minimum presses for 40,629 is:
Which is the smallest three-digit result of exactly 30 presses of +1, +10, and +100?
With only +10,000 and +100, 1,20,500 can be made with:
Practice Problems
- For the +10-only calculator, find the displays after 47, 100, and 435 presses. Can it display 4,357? Explain.
- Represent 321 and 8,300 in three different ways using Chitti’s buttons. Count the presses in each representation.
- Verify both representations of 3,67,813 in the table. Explain why changing 78 hundreds to 78 thousands is not harmless.
- Make 997 using 25 presses and 34 presses. Explain the exchange linking the two routes.
- Find the minimum press count for 56,354 and 66,666 when all needed buttons are present.
- Check the constructions of 102 and 993 using exactly 30 presses. Explain why 100 and 999 fail the remainder condition.
- Use only +10,000 and +100 to make 20,800, 92,100, 1,20,500, 65,30,000, and 70,25,700. Explain which exchanges reduce the count.
- Create a target that +10 can reach but +100 cannot, and another that +100 can reach but +1,000 cannot.
Key Takeaways
• A single addition button reaches multiples of its value from zero. • Different groupings can make the same number with different press counts. • Ten small place-value groups can exchange for one group in the next place. • Digit sum gives the minimum only when every required decimal button is available. • Exactly a given press count and the fewest possible presses are different goals. • Restricted buttons require representations built from the actual available values.