Practice · Build a number from digit conditions in order · Sensim Math

Variant 1 easy 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 $0,1,2,3,4$ 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 $1$ less than the digit in the ones place.
(e) The digit in the thousands place is $0$ .
(f) The value of the digit in the hundred-thousands place is $100{,}000$ .

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute5 carry out the plan

1Hundred-thousands place

#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 \times 100{,}000

Place value just says how many hundred-thousands there are.

2Ones place

#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.

\text{ones digit} = 4

Among 0, 1, 2, 3, 4 the biggest is plainly 4.

3Ten-thousands and hundreds places

#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 = 3

Working backward from the ones digit, '1 less than 4' is 3 for both matching places.

4Thousands place

#7 Identify Subproblems 4.NBT.A.2

The thousands digit is given directly as 0.

\text{thousands digit} = 0

One clue hands this place to us with no work.

5Tens place by 'use every digit'

#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\;\boxed{2}\;4

Every required digit must appear, and the missing one fills the last slot.

Answer: 130,324

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 easy 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 $0,1,2,3,4$ 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 $1$ less than the digit in the ones place.
(e) The digit in the thousands place is $1$ .
(f) The value of the digit in the hundred-thousands place is $200{,}000$ .

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute5 carry out the plan

1Hundred-thousands place

#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 \times 100{,}000

Place value just says how many hundred-thousands there are.

2Ones place

#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.

\text{ones digit} = 4

Among 0, 1, 2, 3, 4 the biggest is plainly 4.

3Ten-thousands and hundreds places

#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 = 3

Working backward from the ones digit, '1 less than 4' is 3 for both matching places.

4Thousands place

#7 Identify Subproblems 4.NBT.A.2

The thousands digit is given directly as 1.

\text{thousands digit} = 1

One clue hands this place to us with no work.

5Tens place by 'use every digit'

#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\;\boxed{0}\;4

Every required digit must appear, and the missing one fills the last slot.

Answer: 231,304

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 easy 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 $0,1,2,3,4$ 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 $1$ less than the digit in the ones place.
(e) The digit in the thousands place is $0$ .
(f) The value of the digit in the hundred-thousands place is $200{,}000$ .

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute5 carry out the plan

1Hundred-thousands place

#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 \times 100{,}000

Place value just says how many hundred-thousands there are.

2Ones place

#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.

\text{ones digit} = 4

Among 0, 1, 2, 3, 4 the biggest is plainly 4.

3Ten-thousands and hundreds places

#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 = 3

Working backward from the ones digit, '1 less than 4' is 3 for both matching places.

4Thousands place

#7 Identify Subproblems 4.NBT.A.2

The thousands digit is given directly as 0.

\text{thousands digit} = 0

One clue hands this place to us with no work.

5Tens place by 'use every digit'

#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\;\boxed{1}\;4

Every required digit must appear, and the missing one fills the last slot.

Answer: 230,314

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 medium 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 $0,1,3,4,5$ 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 $1$ less than the digit in the ones place.
(e) The digit in the thousands place is $0$ .
(f) The value of the digit in the hundred-thousands place is $100{,}000$ .

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute5 carry out the plan

1Hundred-thousands place

#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 \times 100{,}000

Place value just says how many hundred-thousands there are.

2Ones place

#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.

\text{ones digit} = 5

Among 0, 1, 3, 4, 5 the biggest is plainly 5.

3Ten-thousands and hundreds places

#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 = 4

Working backward from the ones digit, '1 less than 5' is 4 for both matching places.

4Thousands place

#7 Identify Subproblems 4.NBT.A.2

The thousands digit is given directly as 0.

\text{thousands digit} = 0

One clue hands this place to us with no work.

5Tens place by 'use every digit'

#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\;\boxed{3}\;5

Every required digit must appear, and the missing one fills the last slot.

Answer: 140,435

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 medium 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 $0,2,3,4,5$ 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 $1$ less than the digit in the ones place.
(e) The digit in the thousands place is $0$ .
(f) The value of the digit in the hundred-thousands place is $200{,}000$ .

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute5 carry out the plan

1Hundred-thousands place

#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 \times 100{,}000

Place value just says how many hundred-thousands there are.

2Ones place

#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.

\text{ones digit} = 5

Among 0, 2, 3, 4, 5 the biggest is plainly 5.

3Ten-thousands and hundreds places

#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 = 4

Working backward from the ones digit, '1 less than 5' is 4 for both matching places.

4Thousands place

#7 Identify Subproblems 4.NBT.A.2

The thousands digit is given directly as 0.

\text{thousands digit} = 0

One clue hands this place to us with no work.

5Tens place by 'use every digit'

#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\;\boxed{3}\;5

Every required digit must appear, and the missing one fills the last slot.

Answer: 240,435

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 medium 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 $0,2,4,5,6$ 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 $1$ less than the digit in the ones place.
(e) The digit in the thousands place is $0$ .
(f) The value of the digit in the hundred-thousands place is $200{,}000$ .

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute5 carry out the plan

1Hundred-thousands place

#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 \times 100{,}000

Place value just says how many hundred-thousands there are.

2Ones place

#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.

\text{ones digit} = 6

Among 0, 2, 4, 5, 6 the biggest is plainly 6.

3Ten-thousands and hundreds places

#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 = 5

Working backward from the ones digit, '1 less than 6' is 5 for both matching places.

4Thousands place

#7 Identify Subproblems 4.NBT.A.2

The thousands digit is given directly as 0.

\text{thousands digit} = 0

One clue hands this place to us with no work.

5Tens place by 'use every digit'

#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\;\boxed{4}\;6

Every required digit must appear, and the missing one fills the last slot.

Answer: 250,546

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 hard 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 $1,2,3,4,5$ 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 $1$ less than the digit in the ones place.
(e) The digit in the thousands place is $1$ .
(f) The value of the digit in the hundred-thousands place is $300{,}000$ .

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute5 carry out the plan

1Hundred-thousands place

#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 \times 100{,}000

Place value just says how many hundred-thousands there are.

2Ones place

#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.

\text{ones digit} = 5

Among 1, 2, 3, 4, 5 the biggest is plainly 5.

3Ten-thousands and hundreds places

#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 = 4

Working backward from the ones digit, '1 less than 5' is 4 for both matching places.

4Thousands place

#7 Identify Subproblems 4.NBT.A.2

The thousands digit is given directly as 1.

\text{thousands digit} = 1

One clue hands this place to us with no work.

5Tens place by 'use every digit'

#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\;\boxed{2}\;5

Every required digit must appear, and the missing one fills the last slot.

Answer: 341,425

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 medium 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 $1,2,3,4,5$ 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 $1$ less than the digit in the ones place.
(e) The digit in the thousands place is $1$ .
(f) The value of the digit in the hundred-thousands place is $200{,}000$ .

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute5 carry out the plan

1Hundred-thousands place

#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 \times 100{,}000

Place value just says how many hundred-thousands there are.

2Ones place

#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.

\text{ones digit} = 5

Among 1, 2, 3, 4, 5 the biggest is plainly 5.

3Ten-thousands and hundreds places

#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 = 4

Working backward from the ones digit, '1 less than 5' is 4 for both matching places.

4Thousands place

#7 Identify Subproblems 4.NBT.A.2

The thousands digit is given directly as 1.

\text{thousands digit} = 1

One clue hands this place to us with no work.

5Tens place by 'use every digit'

#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\;\boxed{3}\;5

Every required digit must appear, and the missing one fills the last slot.

Answer: 241,435

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 hard 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 $1,3,4,5,6$ 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 $1$ less than the digit in the ones place.
(e) The digit in the thousands place is $1$ .
(f) The value of the digit in the hundred-thousands place is $300{,}000$ .

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute5 carry out the plan

1Hundred-thousands place

#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 \times 100{,}000

Place value just says how many hundred-thousands there are.

2Ones place

#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.

\text{ones digit} = 6

Among 1, 3, 4, 5, 6 the biggest is plainly 6.

3Ten-thousands and hundreds places

#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 = 5

Working backward from the ones digit, '1 less than 6' is 5 for both matching places.

4Thousands place

#7 Identify Subproblems 4.NBT.A.2

The thousands digit is given directly as 1.

\text{thousands digit} = 1

One clue hands this place to us with no work.

5Tens place by 'use every digit'

#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\;\boxed{4}\;6

Every required digit must appear, and the missing one fills the last slot.

Answer: 351,546

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 hard 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 $2,3,4,5,6$ 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 $1$ less than the digit in the ones place.
(e) The digit in the thousands place is $3$ .
(f) The value of the digit in the hundred-thousands place is $200{,}000$ .

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute5 carry out the plan

1Hundred-thousands place

#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 \times 100{,}000

Place value just says how many hundred-thousands there are.

2Ones place

#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.

\text{ones digit} = 6

Among 2, 3, 4, 5, 6 the biggest is plainly 6.

3Ten-thousands and hundreds places

#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 = 5

Working backward from the ones digit, '1 less than 6' is 5 for both matching places.

4Thousands place

#7 Identify Subproblems 4.NBT.A.2

The thousands digit is given directly as 3.

\text{thousands digit} = 3

One clue hands this place to us with no work.

5Tens place by 'use every digit'

#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\;\boxed{4}\;6

Every required digit must appear, and the missing one fills the last slot.

Answer: 253,546

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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!

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

Generated variants — 10

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