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Lesson 7 of 9

Large Numbers Around Us · Lesson 7 of 9

Place-value Puzzles and Toothpick Digits

“Use place value and clear game rules to solve digit-card, deletion, and toothpick challenges.”

Learning Objectives

• Construct extreme numbers subject to digit-use, divisibility, and leading-zero rules. • Use place weights to explain digit-swap and number-card puzzles. • Find large retained digit strings while preserving their original order. • Build legal arithmetic expressions using each available number card at most once. • Count and rearrange seven-segment toothpicks while checking every game constraint.

The rules are part of the puzzle

A largest-number puzzle is not solved just by arranging many large digits. You must also obey the allowed digits, the number of copies, the final-digit condition, and whether positions may change. State these rules before comparing candidates, then justify why no better candidate is possible.

When two numbers have the same number of digits, compare from the left. The first place where they differ decides which is larger, because that place is worth more than all the less significant places together. A number written without leading zeros cannot begin with 0.

Example — Use every digit once

Problem
Using each digit 0–9 exactly once, construct the largest multiple of 5 and the smallest even ten-digit number.

  1. 1.A multiple of 5 must end in 0 or 5. Ending in 0 lets the other digits appear in descending order: 9,876,543,210.
  2. 2.If it ends in 5 instead, the descending arrangement of the other digits gives 9,876,432,105, which is smaller at the first differing position. Thus 9,876,543,210 is the largest.
  3. 3.For the smallest even number, begin with 1, the smallest allowed nonzero leading digit, then place 0 next.
  4. 4.Keep the next digits as small as possible: 2, 3, 4, 5, 6, 7. The remaining digits are 8 and 9; the final digit must be even, so put 9 before the final 8.
  5. 5.The smallest result is 1,023,456,798. Every digit is used once, and its final 8 makes it even.

Swapping and deleting digits

Moving a digit changes the place value it receives. A swap changes two positions, while deletion preserves the relative order of all retained digits. These are different operations, so a deletion answer cannot quietly rearrange the surviving digits.

Example — Every swap must make it larger

Problem
Find a nine-digit number for which exchanging any two digits makes a strictly larger number.

  1. 1.If an earlier digit were larger than a later digit, swapping those two would put a smaller digit at the earlier place and make the number smaller.
  2. 2.Equal digits would produce an unchanged number. Therefore all digits must be distinct and strictly increasing from left to right.
  3. 3.Zero cannot occur, because in increasing order it would have to lead. Nine distinct increasing digits from 1–9 give only 123456789.
  4. 4.Swapping any two puts the larger digit into the earlier of the two positions, so the result is larger. Exactly one number satisfies the condition.
Example — Delete ten digits without rearranging

Problem
Delete exactly ten digits from 12345123451234512345 to make the largest possible remaining ten-digit number.

  1. 1.The first retained digit can be selected only from the first eleven positions: enough digits must remain after it to fill the other nine places. The largest available digit there is 5; choose its earliest occurrence, at position 5.
  2. 2.For the second retained digit, another 5 is available at position 10. Choosing it deletes four more digits. Eight digits have now been deleted in total.
  3. 3.The remaining suffix is 1234512345. We must keep eight of these ten digits, so delete its initial 1 and 2 to begin the rest with 3.
  4. 4.The result is 5534512345. It keeps digits in their original order and deletes exactly ten.
  5. 5.Trying to begin with 555 would leave only five digits after the third 5, too few to complete a ten-digit result. Feasibility matters as well as choosing a large digit.
Original groups:1234 5 | 1234 5 | 12 34512345Delete:1234 + 1234 + 12 → 10 digitsKeep:5 | 5 | 34512345Result:5534512345
A deletion keeps order— The highlighted retained groups form 5534512345; the other ten digits are removed.

Two sets of digit cards

Now use two copies of each card 1–9 to fill a seven-digit number and a five-digit number. Not all eighteen cards need to be used, but no digit may occur more than twice across the two numbers. To maximise their sum, assign the largest available digits to the largest place weights across both numbers, rather than filling one number completely first.

Example — Largest sum and smallest difference

Problem
Find the largest sum and the smallest difference of a seven-digit and a five-digit number using at most two copies of each digit 1–9.

  1. 1.For the largest sum, the million and hundred-thousand places exist only in the longer number. Put the two 9s there.
  2. 2.Both numbers have a ten-thousand place, so put the two 8s there. Similarly assign two 7s to thousands, two 6s to hundreds, two 5s to tens, and two 4s to ones.
  3. 3.The numbers are 9,987,654 and 87,654. Their sum is 10,075,308. Assigning a smaller digit to a larger weight while a larger digit occupies a smaller weight could be improved by swapping, which explains the maximum.
  4. 4.For the smallest difference, make the longer number as small as possible: 1,122,334. A seven-digit number always exceeds any five-digit number, so increasing the five-digit number reduces the difference.
  5. 5.The largest available five-digit number is then 99,887. The difference is 1,122,334 − 99,887 = 1,022,447. The digit-copy limits are respected in both constructions.

Why not use both 8s early in the longer number for the sum? One of them would occupy its thousands place while a smaller digit occupied the shorter number’s ten-thousands place. Moving the 8 to the larger weight increases the total. For the difference, the smallest digits belong to the positive place weights and the largest to the subtracted place weights.

Arithmetic with seven number cards

The available cards are 4,000, 13,000, 300, 70,000, 1,50,000, 20, and 5. For each target, use any subset with addition, subtraction, multiplication, or division and brackets, but use each card at most once. A card may be reused in a different target’s expression, not twice inside the same expression.

Compare a candidate with the target by their absolute gap: subtract the smaller from the larger. A gap of zero is an exact solution and cannot be improved. A nonzero gap describes a candidate, not a proof that no closer expression exists; keep searching unless you have a complete argument.

Example — Improve a near answer to an exact one

Problem
Reach 1,10,000 with the available cards.

  1. 1.The candidate 4,000 × (20 + 5) + 13,000 equals 1,13,000, with a gap of 3,000.
  2. 2.Try 13,000 × 20 = 2,60,000, then subtract 1,50,000.
  3. 3.13,000 × 20 − 1,50,000 = 1,10,000, with a gap of zero.
  4. 4.Only the 13,000, 20, and 1,50,000 cards are used, each once. No invented card or repeated card is needed.
TargetLegal expressionResult and gap
2,00,00070,000 × 5 − 1,50,0002,00,000; gap 0
5,80,0004,000 × ((300 × 70,000 ÷ 1,50,000) + 5)5,80,000; gap 0
12,45,000300 × (13,000 + 70,000) ÷ 2012,45,000; gap 0
20,90,80070,000 + ((1,50,000 − 13,000) ÷ 20) × (300 − 5)20,90,750; gap 50; a candidate to try to improve

For the second row, 300 × 70,000 ÷ 1,50,000 = 140, then 140 + 5 = 145 and 4,000 × 145 = 5,80,000. For the last row, the two brackets give 6,850 and 295; their product is 20,20,750, then adding 70,000 gives 20,90,750. The final target remains an open improvement challenge under these operations.

An intermediate result is not an extra card

You may calculate 140 from the given cards, but you cannot introduce a free 4 or 14 just because it makes a target easy. Show how every quantity in an expression comes from an available card. Brackets tell the intended order of operations.

Toothpick digits have their own rules

Make digits using the seven-segment shapes shown below. A stick occupies one full segment, and each completed digit must match one of the displayed forms. Unless a puzzle says otherwise, keep the same positions for an add-or-move challenge and do not allow leading zeros.

06 sticks12 sticks25 sticks35 sticks44 sticks55 sticks66 sticks73 sticks87 sticks96 sticks
Seven-segment digit shapes— Count each segment once; the digit 1 uses two and 8 uses seven.
Digit0123456789
Sticks6255456376
Example — Count and add sticks

Problem
Count the sticks in 5,108 and 42,019, then make 42,019 larger by adding exactly two sticks.

  1. 1.For 5,108, add the digit counts: 5 + 2 + 6 + 7 = 20 sticks.
  2. 2.For 42,019, add 4 + 5 + 6 + 2 + 6 = 23 sticks.
  3. 3.Adding the top segment to its 1 changes it to 7. Adding the lower-left segment to its 9 changes it to 8. The result is 42,078, using two extra sticks.
  4. 4.Another result is 42,879: add a middle segment to 0 to make 8, and a top segment to 1 to make 7. The original sticks remain in place.

Inserting a new digit is a different game. Inserting 1 somewhere in 42019 gives 142019, 412019, 421019, 420119, or 420191; inserting on either side of the original 1 gives the same 420119. Comparing these six-digit results from the left, 421019 is largest.

Example — Move exactly four sticks

Problem
Verify the transformation 63890 → 88078 by moving exactly four sticks.

  1. 1.Change 6 to 8 by adding its missing upper-right segment: one addition.
  2. 2.Change 3 to 8 by adding upper-left and lower-left segments: two more additions.
  3. 3.Change the middle 8 to 0 by removing its middle segment: one removal.
  4. 4.Change 9 to 7 by removing its upper-left, middle, and bottom segments: three more removals.
  5. 5.Change the final 0 to 8 by adding a middle segment: the fourth addition. Four removed sticks supply the four added positions; no sticks are added from outside or discarded.
BeforeAfter30 sticks before and after; exactly 4 moved.
A four-stick move— Four segments are removed from the old number and placed into four new positions.

Exactly twenty-four sticks

When the digit count is not fixed, more digits usually make a larger positive number. Fewer digits usually make a smaller one, provided every stick is used and the first digit is nonzero. We can use these observations to prove extremes rather than guess.

Example — Largest and smallest with 24 sticks

Problem
Using exactly 24 sticks, find the largest and smallest positive whole numbers in these digit shapes.

  1. 1.Every digit needs at least two sticks, so there can be at most twelve digits. Twelve copies of 1 use exactly 24 sticks, giving 111111111111.
  2. 2.Any change to another digit would use more sticks; a number with fewer than twelve digits is smaller. Therefore this is the largest.
  3. 3.At most seven sticks are used by one digit, so three digits can use at most 21. A 24-stick number needs at least four digits.
  4. 4.A leading 1 uses two sticks; even three 8s after it give only 23. The smallest possible leading digit is therefore 2, using five sticks.
  5. 5.For the remaining nineteen sticks, choose 0 next if possible: it leaves thirteen. Another 0 leaves seven, which makes 8. Thus 2008 uses 5 + 6 + 6 + 7 = 24 sticks.
  6. 6.The earliest feasible choices 2, 0, 0, 8 make 2008 the smallest four-digit candidate, and any longer positive number is larger.

Quiz

Quick check

Which arrangement uses 0–9 once and is the largest multiple of 5?

Quick check

The nine-digit number where every two-digit swap makes a larger number is:

Quick check

Which largest sum obeys the two-copies-per-digit rule for a seven-digit and five-digit number?

Quick check

Which expression legally reaches 1,10,000 using the given number cards?

Quick check

How many sticks form 5,108?

Quick check

Inserting 1 into 42019 gives which largest result?

Quick check

The smallest positive whole number using exactly 24 sticks is:

Practice Problems

Practice Problems
  1. Construct the largest multiple of 5 and the smallest even number using 0–9 once each. Explain each earliest digit choice.
  2. Explain why a nine-digit number whose every swap increases it must have distinct increasing digits and cannot contain 0.
  3. Strike out ten digits from 12345123451234512345. Show why your retained result cannot begin with 555.
  4. Fill a seven-digit and a five-digit number from two sets of 1–9 to maximise their sum. Compare the place weights across both numbers. Then minimise their difference.
  5. Use the seven number cards to reach 1,10,000, 2,00,000, 5,80,000, and 12,45,000 exactly. Check each card is used at most once.
  6. Try to improve the gap-50 candidate for 20,90,800. State your operations and calculate the gap without claiming optimality unless you can justify it.
  7. Make 42,019 with 23 sticks. Add exactly two sticks to produce two different larger numbers, keeping every original segment.
  8. List all results of inserting 1 into 42019 and identify repeated results. Explain which is largest.
  9. Draw 63890 → 88078 and label the four removed and four added segments. Find another legal four-stick move if possible.
  10. Prove the maximum and minimum with exactly 24 sticks. Then create your own toothpick challenge with clear rules.

Key Takeaways

Key Takeaways

• Extreme-number puzzles require both place-value reasoning and every stated constraint. • At equal digit lengths, the first differing position decides the comparison. • Deleting digits preserves order; exchanging or inserting digits follows different rules. • For a largest sum, allocate large cards to the largest weights across both numbers. • Use each available number card at most once per target and report a candidate’s gap honestly. • Toothpick moves conserve sticks; added-stick puzzles keep the original segments. • Seven-segment stick counts and no-leading-zero rules help prove number extremes.

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Large Numbers in Facts and Thought Experiments

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Counting Digits and Exploring Number Names