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

Variant 1 easy

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 $100{,}000$ .

Your answer

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

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 $200{,}000$ .

Your answer

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

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 $200{,}000$ .

Your answer

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

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 $100{,}000$ .

Your answer

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

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 $200{,}000$ .

Your answer

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

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 $200{,}000$ .

Your answer

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

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 $300{,}000$ .

Your answer

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

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 $200{,}000$ .

Your answer

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

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 $300{,}000$ .

Your answer

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

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 $200{,}000$ .

Your answer

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

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Where next?