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← 4-1 · Build a number from digit conditions in order · Pin Down a Number from Digit and Range Conditions

Build a number from digit conditions in order · 10 practice problems

4.NBT.A.2

Generated variants — 10

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 answer: 130,324

Find the number that satisfies all of the following conditions.

  • (a) It is a six-digit number.
  • (b) Each of the digits 0,1,2,3,40,1,2,3,4 is used.
  • (c) The largest digit is in the lowest place (the ones place).
  • (d) The digit in the ten-thousands place equals the digit in the hundreds place, and it is 11 less than the digit in the ones place.
  • (e) The digit in the thousands place is 00.
  • (f) The value of the digit in the hundred-thousands place is 100,000100{,}000.
Show solution

Understand

Build a single six-digit number whose six place-value digits satisfy a list of clues, using each of the digits 0, 1, 2, 3, 4 at least once.

Givens
  • The number has six digits (hundred-thousands down to ones).
  • Each of the digits 0, 1, 2, 3, 4 is used.
  • The largest digit sits in the ones place.
  • The ten-thousands digit equals the hundreds digit, and it is 1 less than the ones digit.
  • The thousands digit is 0.
  • The hundred-thousands digit has place value 100{,}000.
Unknowns
  • The six-digit number that satisfies every condition
Constraints
  • Only the digits 0, 1, 2, 3, 4 may appear, and all five must show up at least once.
  • A six-digit number cannot start with 0.

Plan

#6 Guess and Check · also uses: #7 Identify Subproblems#11 Work Backwards

Each clue pins down one place, so I fix the places I know for certain first (a subproblem per place), then use the 'use every digit' rule to fill the one leftover place, checking the result against all clues.

Execute

#11 Work Backwards 4.NBT.A.2
A place value of 100{,}000 in the hundred-thousands place means that digit is 1.
100,000=1×100,000100{,}000 = 1 \times 100{,}000
Place value just says how many hundred-thousands there are.
#6 Guess and Check 4.NBT.A.2
The largest of the allowed digits 0, 1, 2, 3, 4 is 4, and it goes in the ones place.
ones digit=4\text{ones digit} = 4
Among 0, 1, 2, 3, 4 the biggest is plainly 4.
#11 Work Backwards 4.NBT.A.2
These two digits are equal and each is 1 less than the ones digit 4, so both are 3.
4−1=34 - 1 = 3
Working backward from the ones digit, '1 less than 4' is 3 for both matching places.
#7 Identify Subproblems 4.NBT.A.2
The thousands digit is given directly as 0.
thousands digit=0\text{thousands digit} = 0
One clue hands this place to us with no work.
#6 Guess and Check 4.NBT.A.2
So far the placed digits are 1, 3, 0, 3, ?, 4. The digit 2 has not yet been used, so the only empty place (tens) must be 2.
1  3  0  3  2  41\;3\;0\;3\;\boxed{2}\;4
Every required digit must appear, and the missing one fills the last slot.
Answer: 130,324

Review

130,324 is a six-digit number using only 0, 1, 2, 3, 4 with all five present; its ones digit 4 is the largest, ten-thousands and hundreds are both 3 (which is 4-1), thousands is 0, and the lead digit 1 gives place value 100{,}000. Every clue holds.

Make a systematic list (tool 2): write the place names in a row, fill each from its clue, and the single blank that remains forces the missing digit.

Standards · min grade 4

  • 4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Reading each place value and assembling the six-digit number from the digit clues.
💡 This only needs Grade 4 place-value sense: pin down each digit from its clue, then fill the last spot with the digit you haven't used yet!
Variant 2 answer: 351,546

Find the number that satisfies all of the following conditions.

  • (a) It is a six-digit number.
  • (b) Each of the digits 1,3,4,5,61,3,4,5,6 is used.
  • (c) The largest digit is in the lowest place (the ones place).
  • (d) The digit in the ten-thousands place equals the digit in the hundreds place, and it is 11 less than the digit in the ones place.
  • (e) The digit in the thousands place is 11.
  • (f) The value of the digit in the hundred-thousands place is 300,000300{,}000.
Show solution

Understand

Build a single six-digit number whose six place-value digits satisfy a list of clues, using each of the digits 1, 3, 4, 5, 6 at least once.

Givens
  • The number has six digits (hundred-thousands down to ones).
  • Each of the digits 1, 3, 4, 5, 6 is used.
  • The largest digit sits in the ones place.
  • The ten-thousands digit equals the hundreds digit, and it is 1 less than the ones digit.
  • The thousands digit is 1.
  • The hundred-thousands digit has place value 300{,}000.
Unknowns
  • The six-digit number that satisfies every condition
Constraints
  • Only the digits 1, 3, 4, 5, 6 may appear, and all five must show up at least once.
  • A six-digit number cannot start with 0.

Plan

#6 Guess and Check · also uses: #7 Identify Subproblems#11 Work Backwards

Each clue pins down one place, so I fix the places I know for certain first (a subproblem per place), then use the 'use every digit' rule to fill the one leftover place, checking the result against all clues.

Execute

#11 Work Backwards 4.NBT.A.2
A place value of 300{,}000 in the hundred-thousands place means that digit is 3.
300,000=3×100,000300{,}000 = 3 \times 100{,}000
Place value just says how many hundred-thousands there are.
#6 Guess and Check 4.NBT.A.2
The largest of the allowed digits 1, 3, 4, 5, 6 is 6, and it goes in the ones place.
ones digit=6\text{ones digit} = 6
Among 1, 3, 4, 5, 6 the biggest is plainly 6.
#11 Work Backwards 4.NBT.A.2
These two digits are equal and each is 1 less than the ones digit 6, so both are 5.
6−1=56 - 1 = 5
Working backward from the ones digit, '1 less than 6' is 5 for both matching places.
#7 Identify Subproblems 4.NBT.A.2
The thousands digit is given directly as 1.
thousands digit=1\text{thousands digit} = 1
One clue hands this place to us with no work.
#6 Guess and Check 4.NBT.A.2
So far the placed digits are 3, 5, 1, 5, ?, 6. The digit 4 has not yet been used, so the only empty place (tens) must be 4.
3  5  1  5  4  63\;5\;1\;5\;\boxed{4}\;6
Every required digit must appear, and the missing one fills the last slot.
Answer: 351,546

Review

351,546 is a six-digit number using only 1, 3, 4, 5, 6 with all five present; its ones digit 6 is the largest, ten-thousands and hundreds are both 5 (which is 6-1), thousands is 1, and the lead digit 3 gives place value 300{,}000. Every clue holds.

Make a systematic list (tool 2): write the place names in a row, fill each from its clue, and the single blank that remains forces the missing digit.

Standards · min grade 4

  • 4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Reading each place value and assembling the six-digit number from the digit clues.
💡 This only needs Grade 4 place-value sense: pin down each digit from its clue, then fill the last spot with the digit you haven't used yet!
Variant 3 answer: 240,435

Find the number that satisfies all of the following conditions.

  • (a) It is a six-digit number.
  • (b) Each of the digits 0,2,3,4,50,2,3,4,5 is used.
  • (c) The largest digit is in the lowest place (the ones place).
  • (d) The digit in the ten-thousands place equals the digit in the hundreds place, and it is 11 less than the digit in the ones place.
  • (e) The digit in the thousands place is 00.
  • (f) The value of the digit in the hundred-thousands place is 200,000200{,}000.
Show solution

Understand

Build a single six-digit number whose six place-value digits satisfy a list of clues, using each of the digits 0, 2, 3, 4, 5 at least once.

Givens
  • The number has six digits (hundred-thousands down to ones).
  • Each of the digits 0, 2, 3, 4, 5 is used.
  • The largest digit sits in the ones place.
  • The ten-thousands digit equals the hundreds digit, and it is 1 less than the ones digit.
  • The thousands digit is 0.
  • The hundred-thousands digit has place value 200{,}000.
Unknowns
  • The six-digit number that satisfies every condition
Constraints
  • Only the digits 0, 2, 3, 4, 5 may appear, and all five must show up at least once.
  • A six-digit number cannot start with 0.

Plan

#6 Guess and Check · also uses: #7 Identify Subproblems#11 Work Backwards

Each clue pins down one place, so I fix the places I know for certain first (a subproblem per place), then use the 'use every digit' rule to fill the one leftover place, checking the result against all clues.

Execute

#11 Work Backwards 4.NBT.A.2
A place value of 200{,}000 in the hundred-thousands place means that digit is 2.
200,000=2×100,000200{,}000 = 2 \times 100{,}000
Place value just says how many hundred-thousands there are.
#6 Guess and Check 4.NBT.A.2
The largest of the allowed digits 0, 2, 3, 4, 5 is 5, and it goes in the ones place.
ones digit=5\text{ones digit} = 5
Among 0, 2, 3, 4, 5 the biggest is plainly 5.
#11 Work Backwards 4.NBT.A.2
These two digits are equal and each is 1 less than the ones digit 5, so both are 4.
5−1=45 - 1 = 4
Working backward from the ones digit, '1 less than 5' is 4 for both matching places.
#7 Identify Subproblems 4.NBT.A.2
The thousands digit is given directly as 0.
thousands digit=0\text{thousands digit} = 0
One clue hands this place to us with no work.
#6 Guess and Check 4.NBT.A.2
So far the placed digits are 2, 4, 0, 4, ?, 5. The digit 3 has not yet been used, so the only empty place (tens) must be 3.
2  4  0  4  3  52\;4\;0\;4\;\boxed{3}\;5
Every required digit must appear, and the missing one fills the last slot.
Answer: 240,435

Review

240,435 is a six-digit number using only 0, 2, 3, 4, 5 with all five present; its ones digit 5 is the largest, ten-thousands and hundreds are both 4 (which is 5-1), thousands is 0, and the lead digit 2 gives place value 200{,}000. Every clue holds.

Make a systematic list (tool 2): write the place names in a row, fill each from its clue, and the single blank that remains forces the missing digit.

Standards · min grade 4

  • 4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Reading each place value and assembling the six-digit number from the digit clues.
💡 This only needs Grade 4 place-value sense: pin down each digit from its clue, then fill the last spot with the digit you haven't used yet!
Variant 4 answer: 341,425

Find the number that satisfies all of the following conditions.

  • (a) It is a six-digit number.
  • (b) Each of the digits 1,2,3,4,51,2,3,4,5 is used.
  • (c) The largest digit is in the lowest place (the ones place).
  • (d) The digit in the ten-thousands place equals the digit in the hundreds place, and it is 11 less than the digit in the ones place.
  • (e) The digit in the thousands place is 11.
  • (f) The value of the digit in the hundred-thousands place is 300,000300{,}000.
Show solution

Understand

Build a single six-digit number whose six place-value digits satisfy a list of clues, using each of the digits 1, 2, 3, 4, 5 at least once.

Givens
  • The number has six digits (hundred-thousands down to ones).
  • Each of the digits 1, 2, 3, 4, 5 is used.
  • The largest digit sits in the ones place.
  • The ten-thousands digit equals the hundreds digit, and it is 1 less than the ones digit.
  • The thousands digit is 1.
  • The hundred-thousands digit has place value 300{,}000.
Unknowns
  • The six-digit number that satisfies every condition
Constraints
  • Only the digits 1, 2, 3, 4, 5 may appear, and all five must show up at least once.
  • A six-digit number cannot start with 0.

Plan

#6 Guess and Check · also uses: #7 Identify Subproblems#11 Work Backwards

Each clue pins down one place, so I fix the places I know for certain first (a subproblem per place), then use the 'use every digit' rule to fill the one leftover place, checking the result against all clues.

Execute

#11 Work Backwards 4.NBT.A.2
A place value of 300{,}000 in the hundred-thousands place means that digit is 3.
300,000=3×100,000300{,}000 = 3 \times 100{,}000
Place value just says how many hundred-thousands there are.
#6 Guess and Check 4.NBT.A.2
The largest of the allowed digits 1, 2, 3, 4, 5 is 5, and it goes in the ones place.
ones digit=5\text{ones digit} = 5
Among 1, 2, 3, 4, 5 the biggest is plainly 5.
#11 Work Backwards 4.NBT.A.2
These two digits are equal and each is 1 less than the ones digit 5, so both are 4.
5−1=45 - 1 = 4
Working backward from the ones digit, '1 less than 5' is 4 for both matching places.
#7 Identify Subproblems 4.NBT.A.2
The thousands digit is given directly as 1.
thousands digit=1\text{thousands digit} = 1
One clue hands this place to us with no work.
#6 Guess and Check 4.NBT.A.2
So far the placed digits are 3, 4, 1, 4, ?, 5. The digit 2 has not yet been used, so the only empty place (tens) must be 2.
3  4  1  4  2  53\;4\;1\;4\;\boxed{2}\;5
Every required digit must appear, and the missing one fills the last slot.
Answer: 341,425

Review

341,425 is a six-digit number using only 1, 2, 3, 4, 5 with all five present; its ones digit 5 is the largest, ten-thousands and hundreds are both 4 (which is 5-1), thousands is 1, and the lead digit 3 gives place value 300{,}000. Every clue holds.

Make a systematic list (tool 2): write the place names in a row, fill each from its clue, and the single blank that remains forces the missing digit.

Standards · min grade 4

  • 4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Reading each place value and assembling the six-digit number from the digit clues.
💡 This only needs Grade 4 place-value sense: pin down each digit from its clue, then fill the last spot with the digit you haven't used yet!
Variant 5 answer: 253,546

Find the number that satisfies all of the following conditions.

  • (a) It is a six-digit number.
  • (b) Each of the digits 2,3,4,5,62,3,4,5,6 is used.
  • (c) The largest digit is in the lowest place (the ones place).
  • (d) The digit in the ten-thousands place equals the digit in the hundreds place, and it is 11 less than the digit in the ones place.
  • (e) The digit in the thousands place is 33.
  • (f) The value of the digit in the hundred-thousands place is 200,000200{,}000.
Show solution

Understand

Build a single six-digit number whose six place-value digits satisfy a list of clues, using each of the digits 2, 3, 4, 5, 6 at least once.

Givens
  • The number has six digits (hundred-thousands down to ones).
  • Each of the digits 2, 3, 4, 5, 6 is used.
  • The largest digit sits in the ones place.
  • The ten-thousands digit equals the hundreds digit, and it is 1 less than the ones digit.
  • The thousands digit is 3.
  • The hundred-thousands digit has place value 200{,}000.
Unknowns
  • The six-digit number that satisfies every condition
Constraints
  • Only the digits 2, 3, 4, 5, 6 may appear, and all five must show up at least once.
  • A six-digit number cannot start with 0.

Plan

#6 Guess and Check · also uses: #7 Identify Subproblems#11 Work Backwards

Each clue pins down one place, so I fix the places I know for certain first (a subproblem per place), then use the 'use every digit' rule to fill the one leftover place, checking the result against all clues.

Execute

#11 Work Backwards 4.NBT.A.2
A place value of 200{,}000 in the hundred-thousands place means that digit is 2.
200,000=2×100,000200{,}000 = 2 \times 100{,}000
Place value just says how many hundred-thousands there are.
#6 Guess and Check 4.NBT.A.2
The largest of the allowed digits 2, 3, 4, 5, 6 is 6, and it goes in the ones place.
ones digit=6\text{ones digit} = 6
Among 2, 3, 4, 5, 6 the biggest is plainly 6.
#11 Work Backwards 4.NBT.A.2
These two digits are equal and each is 1 less than the ones digit 6, so both are 5.
6−1=56 - 1 = 5
Working backward from the ones digit, '1 less than 6' is 5 for both matching places.
#7 Identify Subproblems 4.NBT.A.2
The thousands digit is given directly as 3.
thousands digit=3\text{thousands digit} = 3
One clue hands this place to us with no work.
#6 Guess and Check 4.NBT.A.2
So far the placed digits are 2, 5, 3, 5, ?, 6. The digit 4 has not yet been used, so the only empty place (tens) must be 4.
2  5  3  5  4  62\;5\;3\;5\;\boxed{4}\;6
Every required digit must appear, and the missing one fills the last slot.
Answer: 253,546

Review

253,546 is a six-digit number using only 2, 3, 4, 5, 6 with all five present; its ones digit 6 is the largest, ten-thousands and hundreds are both 5 (which is 6-1), thousands is 3, and the lead digit 2 gives place value 200{,}000. Every clue holds.

Make a systematic list (tool 2): write the place names in a row, fill each from its clue, and the single blank that remains forces the missing digit.

Standards · min grade 4

  • 4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Reading each place value and assembling the six-digit number from the digit clues.
💡 This only needs Grade 4 place-value sense: pin down each digit from its clue, then fill the last spot with the digit you haven't used yet!
Variant 6 answer: 231,304

Find the number that satisfies all of the following conditions.

  • (a) It is a six-digit number.
  • (b) Each of the digits 0,1,2,3,40,1,2,3,4 is used.
  • (c) The largest digit is in the lowest place (the ones place).
  • (d) The digit in the ten-thousands place equals the digit in the hundreds place, and it is 11 less than the digit in the ones place.
  • (e) The digit in the thousands place is 11.
  • (f) The value of the digit in the hundred-thousands place is 200,000200{,}000.
Show solution

Understand

Build a single six-digit number whose six place-value digits satisfy a list of clues, using each of the digits 0, 1, 2, 3, 4 at least once.

Givens
  • The number has six digits (hundred-thousands down to ones).
  • Each of the digits 0, 1, 2, 3, 4 is used.
  • The largest digit sits in the ones place.
  • The ten-thousands digit equals the hundreds digit, and it is 1 less than the ones digit.
  • The thousands digit is 1.
  • The hundred-thousands digit has place value 200{,}000.
Unknowns
  • The six-digit number that satisfies every condition
Constraints
  • Only the digits 0, 1, 2, 3, 4 may appear, and all five must show up at least once.
  • A six-digit number cannot start with 0.

Plan

#6 Guess and Check · also uses: #7 Identify Subproblems#11 Work Backwards

Each clue pins down one place, so I fix the places I know for certain first (a subproblem per place), then use the 'use every digit' rule to fill the one leftover place, checking the result against all clues.

Execute

#11 Work Backwards 4.NBT.A.2
A place value of 200{,}000 in the hundred-thousands place means that digit is 2.
200,000=2×100,000200{,}000 = 2 \times 100{,}000
Place value just says how many hundred-thousands there are.
#6 Guess and Check 4.NBT.A.2
The largest of the allowed digits 0, 1, 2, 3, 4 is 4, and it goes in the ones place.
ones digit=4\text{ones digit} = 4
Among 0, 1, 2, 3, 4 the biggest is plainly 4.
#11 Work Backwards 4.NBT.A.2
These two digits are equal and each is 1 less than the ones digit 4, so both are 3.
4−1=34 - 1 = 3
Working backward from the ones digit, '1 less than 4' is 3 for both matching places.
#7 Identify Subproblems 4.NBT.A.2
The thousands digit is given directly as 1.
thousands digit=1\text{thousands digit} = 1
One clue hands this place to us with no work.
#6 Guess and Check 4.NBT.A.2
So far the placed digits are 2, 3, 1, 3, ?, 4. The digit 0 has not yet been used, so the only empty place (tens) must be 0.
2  3  1  3  0  42\;3\;1\;3\;\boxed{0}\;4
Every required digit must appear, and the missing one fills the last slot.
Answer: 231,304

Review

231,304 is a six-digit number using only 0, 1, 2, 3, 4 with all five present; its ones digit 4 is the largest, ten-thousands and hundreds are both 3 (which is 4-1), thousands is 1, and the lead digit 2 gives place value 200{,}000. Every clue holds.

Make a systematic list (tool 2): write the place names in a row, fill each from its clue, and the single blank that remains forces the missing digit.

Standards · min grade 4

  • 4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Reading each place value and assembling the six-digit number from the digit clues.
💡 This only needs Grade 4 place-value sense: pin down each digit from its clue, then fill the last spot with the digit you haven't used yet!
Variant 7 answer: 140,435

Find the number that satisfies all of the following conditions.

  • (a) It is a six-digit number.
  • (b) Each of the digits 0,1,3,4,50,1,3,4,5 is used.
  • (c) The largest digit is in the lowest place (the ones place).
  • (d) The digit in the ten-thousands place equals the digit in the hundreds place, and it is 11 less than the digit in the ones place.
  • (e) The digit in the thousands place is 00.
  • (f) The value of the digit in the hundred-thousands place is 100,000100{,}000.
Show solution

Understand

Build a single six-digit number whose six place-value digits satisfy a list of clues, using each of the digits 0, 1, 3, 4, 5 at least once.

Givens
  • The number has six digits (hundred-thousands down to ones).
  • Each of the digits 0, 1, 3, 4, 5 is used.
  • The largest digit sits in the ones place.
  • The ten-thousands digit equals the hundreds digit, and it is 1 less than the ones digit.
  • The thousands digit is 0.
  • The hundred-thousands digit has place value 100{,}000.
Unknowns
  • The six-digit number that satisfies every condition
Constraints
  • Only the digits 0, 1, 3, 4, 5 may appear, and all five must show up at least once.
  • A six-digit number cannot start with 0.

Plan

#6 Guess and Check · also uses: #7 Identify Subproblems#11 Work Backwards

Each clue pins down one place, so I fix the places I know for certain first (a subproblem per place), then use the 'use every digit' rule to fill the one leftover place, checking the result against all clues.

Execute

#11 Work Backwards 4.NBT.A.2
A place value of 100{,}000 in the hundred-thousands place means that digit is 1.
100,000=1×100,000100{,}000 = 1 \times 100{,}000
Place value just says how many hundred-thousands there are.
#6 Guess and Check 4.NBT.A.2
The largest of the allowed digits 0, 1, 3, 4, 5 is 5, and it goes in the ones place.
ones digit=5\text{ones digit} = 5
Among 0, 1, 3, 4, 5 the biggest is plainly 5.
#11 Work Backwards 4.NBT.A.2
These two digits are equal and each is 1 less than the ones digit 5, so both are 4.
5−1=45 - 1 = 4
Working backward from the ones digit, '1 less than 5' is 4 for both matching places.
#7 Identify Subproblems 4.NBT.A.2
The thousands digit is given directly as 0.
thousands digit=0\text{thousands digit} = 0
One clue hands this place to us with no work.
#6 Guess and Check 4.NBT.A.2
So far the placed digits are 1, 4, 0, 4, ?, 5. The digit 3 has not yet been used, so the only empty place (tens) must be 3.
1  4  0  4  3  51\;4\;0\;4\;\boxed{3}\;5
Every required digit must appear, and the missing one fills the last slot.
Answer: 140,435

Review

140,435 is a six-digit number using only 0, 1, 3, 4, 5 with all five present; its ones digit 5 is the largest, ten-thousands and hundreds are both 4 (which is 5-1), thousands is 0, and the lead digit 1 gives place value 100{,}000. Every clue holds.

Make a systematic list (tool 2): write the place names in a row, fill each from its clue, and the single blank that remains forces the missing digit.

Standards · min grade 4

  • 4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Reading each place value and assembling the six-digit number from the digit clues.
💡 This only needs Grade 4 place-value sense: pin down each digit from its clue, then fill the last spot with the digit you haven't used yet!
Variant 8 answer: 230,314

Find the number that satisfies all of the following conditions.

  • (a) It is a six-digit number.
  • (b) Each of the digits 0,1,2,3,40,1,2,3,4 is used.
  • (c) The largest digit is in the lowest place (the ones place).
  • (d) The digit in the ten-thousands place equals the digit in the hundreds place, and it is 11 less than the digit in the ones place.
  • (e) The digit in the thousands place is 00.
  • (f) The value of the digit in the hundred-thousands place is 200,000200{,}000.
Show solution

Understand

Build a single six-digit number whose six place-value digits satisfy a list of clues, using each of the digits 0, 1, 2, 3, 4 at least once.

Givens
  • The number has six digits (hundred-thousands down to ones).
  • Each of the digits 0, 1, 2, 3, 4 is used.
  • The largest digit sits in the ones place.
  • The ten-thousands digit equals the hundreds digit, and it is 1 less than the ones digit.
  • The thousands digit is 0.
  • The hundred-thousands digit has place value 200{,}000.
Unknowns
  • The six-digit number that satisfies every condition
Constraints
  • Only the digits 0, 1, 2, 3, 4 may appear, and all five must show up at least once.
  • A six-digit number cannot start with 0.

Plan

#6 Guess and Check · also uses: #7 Identify Subproblems#11 Work Backwards

Each clue pins down one place, so I fix the places I know for certain first (a subproblem per place), then use the 'use every digit' rule to fill the one leftover place, checking the result against all clues.

Execute

#11 Work Backwards 4.NBT.A.2
A place value of 200{,}000 in the hundred-thousands place means that digit is 2.
200,000=2×100,000200{,}000 = 2 \times 100{,}000
Place value just says how many hundred-thousands there are.
#6 Guess and Check 4.NBT.A.2
The largest of the allowed digits 0, 1, 2, 3, 4 is 4, and it goes in the ones place.
ones digit=4\text{ones digit} = 4
Among 0, 1, 2, 3, 4 the biggest is plainly 4.
#11 Work Backwards 4.NBT.A.2
These two digits are equal and each is 1 less than the ones digit 4, so both are 3.
4−1=34 - 1 = 3
Working backward from the ones digit, '1 less than 4' is 3 for both matching places.
#7 Identify Subproblems 4.NBT.A.2
The thousands digit is given directly as 0.
thousands digit=0\text{thousands digit} = 0
One clue hands this place to us with no work.
#6 Guess and Check 4.NBT.A.2
So far the placed digits are 2, 3, 0, 3, ?, 4. The digit 1 has not yet been used, so the only empty place (tens) must be 1.
2  3  0  3  1  42\;3\;0\;3\;\boxed{1}\;4
Every required digit must appear, and the missing one fills the last slot.
Answer: 230,314

Review

230,314 is a six-digit number using only 0, 1, 2, 3, 4 with all five present; its ones digit 4 is the largest, ten-thousands and hundreds are both 3 (which is 4-1), thousands is 0, and the lead digit 2 gives place value 200{,}000. Every clue holds.

Make a systematic list (tool 2): write the place names in a row, fill each from its clue, and the single blank that remains forces the missing digit.

Standards · min grade 4

  • 4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Reading each place value and assembling the six-digit number from the digit clues.
💡 This only needs Grade 4 place-value sense: pin down each digit from its clue, then fill the last spot with the digit you haven't used yet!
Variant 9 answer: 241,435

Find the number that satisfies all of the following conditions.

  • (a) It is a six-digit number.
  • (b) Each of the digits 1,2,3,4,51,2,3,4,5 is used.
  • (c) The largest digit is in the lowest place (the ones place).
  • (d) The digit in the ten-thousands place equals the digit in the hundreds place, and it is 11 less than the digit in the ones place.
  • (e) The digit in the thousands place is 11.
  • (f) The value of the digit in the hundred-thousands place is 200,000200{,}000.
Show solution

Understand

Build a single six-digit number whose six place-value digits satisfy a list of clues, using each of the digits 1, 2, 3, 4, 5 at least once.

Givens
  • The number has six digits (hundred-thousands down to ones).
  • Each of the digits 1, 2, 3, 4, 5 is used.
  • The largest digit sits in the ones place.
  • The ten-thousands digit equals the hundreds digit, and it is 1 less than the ones digit.
  • The thousands digit is 1.
  • The hundred-thousands digit has place value 200{,}000.
Unknowns
  • The six-digit number that satisfies every condition
Constraints
  • Only the digits 1, 2, 3, 4, 5 may appear, and all five must show up at least once.
  • A six-digit number cannot start with 0.

Plan

#6 Guess and Check · also uses: #7 Identify Subproblems#11 Work Backwards

Each clue pins down one place, so I fix the places I know for certain first (a subproblem per place), then use the 'use every digit' rule to fill the one leftover place, checking the result against all clues.

Execute

#11 Work Backwards 4.NBT.A.2
A place value of 200{,}000 in the hundred-thousands place means that digit is 2.
200,000=2×100,000200{,}000 = 2 \times 100{,}000
Place value just says how many hundred-thousands there are.
#6 Guess and Check 4.NBT.A.2
The largest of the allowed digits 1, 2, 3, 4, 5 is 5, and it goes in the ones place.
ones digit=5\text{ones digit} = 5
Among 1, 2, 3, 4, 5 the biggest is plainly 5.
#11 Work Backwards 4.NBT.A.2
These two digits are equal and each is 1 less than the ones digit 5, so both are 4.
5−1=45 - 1 = 4
Working backward from the ones digit, '1 less than 5' is 4 for both matching places.
#7 Identify Subproblems 4.NBT.A.2
The thousands digit is given directly as 1.
thousands digit=1\text{thousands digit} = 1
One clue hands this place to us with no work.
#6 Guess and Check 4.NBT.A.2
So far the placed digits are 2, 4, 1, 4, ?, 5. The digit 3 has not yet been used, so the only empty place (tens) must be 3.
2  4  1  4  3  52\;4\;1\;4\;\boxed{3}\;5
Every required digit must appear, and the missing one fills the last slot.
Answer: 241,435

Review

241,435 is a six-digit number using only 1, 2, 3, 4, 5 with all five present; its ones digit 5 is the largest, ten-thousands and hundreds are both 4 (which is 5-1), thousands is 1, and the lead digit 2 gives place value 200{,}000. Every clue holds.

Make a systematic list (tool 2): write the place names in a row, fill each from its clue, and the single blank that remains forces the missing digit.

Standards · min grade 4

  • 4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Reading each place value and assembling the six-digit number from the digit clues.
💡 This only needs Grade 4 place-value sense: pin down each digit from its clue, then fill the last spot with the digit you haven't used yet!
Variant 10 answer: 250,546

Find the number that satisfies all of the following conditions.

  • (a) It is a six-digit number.
  • (b) Each of the digits 0,2,4,5,60,2,4,5,6 is used.
  • (c) The largest digit is in the lowest place (the ones place).
  • (d) The digit in the ten-thousands place equals the digit in the hundreds place, and it is 11 less than the digit in the ones place.
  • (e) The digit in the thousands place is 00.
  • (f) The value of the digit in the hundred-thousands place is 200,000200{,}000.
Show solution

Understand

Build a single six-digit number whose six place-value digits satisfy a list of clues, using each of the digits 0, 2, 4, 5, 6 at least once.

Givens
  • The number has six digits (hundred-thousands down to ones).
  • Each of the digits 0, 2, 4, 5, 6 is used.
  • The largest digit sits in the ones place.
  • The ten-thousands digit equals the hundreds digit, and it is 1 less than the ones digit.
  • The thousands digit is 0.
  • The hundred-thousands digit has place value 200{,}000.
Unknowns
  • The six-digit number that satisfies every condition
Constraints
  • Only the digits 0, 2, 4, 5, 6 may appear, and all five must show up at least once.
  • A six-digit number cannot start with 0.

Plan

#6 Guess and Check · also uses: #7 Identify Subproblems#11 Work Backwards

Each clue pins down one place, so I fix the places I know for certain first (a subproblem per place), then use the 'use every digit' rule to fill the one leftover place, checking the result against all clues.

Execute

#11 Work Backwards 4.NBT.A.2
A place value of 200{,}000 in the hundred-thousands place means that digit is 2.
200,000=2×100,000200{,}000 = 2 \times 100{,}000
Place value just says how many hundred-thousands there are.
#6 Guess and Check 4.NBT.A.2
The largest of the allowed digits 0, 2, 4, 5, 6 is 6, and it goes in the ones place.
ones digit=6\text{ones digit} = 6
Among 0, 2, 4, 5, 6 the biggest is plainly 6.
#11 Work Backwards 4.NBT.A.2
These two digits are equal and each is 1 less than the ones digit 6, so both are 5.
6−1=56 - 1 = 5
Working backward from the ones digit, '1 less than 6' is 5 for both matching places.
#7 Identify Subproblems 4.NBT.A.2
The thousands digit is given directly as 0.
thousands digit=0\text{thousands digit} = 0
One clue hands this place to us with no work.
#6 Guess and Check 4.NBT.A.2
So far the placed digits are 2, 5, 0, 5, ?, 6. The digit 4 has not yet been used, so the only empty place (tens) must be 4.
2  5  0  5  4  62\;5\;0\;5\;\boxed{4}\;6
Every required digit must appear, and the missing one fills the last slot.
Answer: 250,546

Review

250,546 is a six-digit number using only 0, 2, 4, 5, 6 with all five present; its ones digit 6 is the largest, ten-thousands and hundreds are both 5 (which is 6-1), thousands is 0, and the lead digit 2 gives place value 200{,}000. Every clue holds.

Make a systematic list (tool 2): write the place names in a row, fill each from its clue, and the single blank that remains forces the missing digit.

Standards · min grade 4

  • 4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Reading each place value and assembling the six-digit number from the digit clues.
💡 This only needs Grade 4 place-value sense: pin down each digit from its clue, then fill the last spot with the digit you haven't used yet!