Depth · one concept from first principles · step 2 of 4

Build a number from digit conditions in order

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 $200{,}000$ .

Base-ten numbers

Your answer

How to solve

Strategy Guess and Check — 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.

1STEP 1

Hundred-thousands place

A place value of 200,000 in the hundred-thousands place means that digit is 2.

200,000 = 2 × 100,000

2STEP 2

Ones place

The largest of the allowed digits 0,1,2,3,4 is 4, and it goes in the ones place.

ones digit = 4

3STEP 3

Ten-thousands and hundreds places

These two digits are equal and each is 1 less than the ones digit 4, so both are 3.

4 - 1 = 3

4STEP 4

Thousands place

The thousands digit is given directly as 0.

thousands digit = 0

5STEP 5

Tens place by 'use every digit'

Digits 0, 2, 3, 4 are placed; only 1 is still unused, so the last empty place (tens) is 1.

2 3 0 3 1 4

Answer

230,314

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.

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!

Hundred-thousands place
Ones place
Ten-thousands and hundreds places
Thousands place
Tens place by 'use every digit'

Where next?

Another one like thissuggested

10 practice problems

Same look, different logicsuggested

Ten of a unit carries to the next place

Step back and solidifysuggested

Find a number satisfying product bounds Bigger addend, bigger sum

16 reasoning problems grow out of this type — the bridge toward competition →

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▶ Practice — 10 problems

The same shape shows up later on the AMC, as Branching State Enumeration, Casework by Position / Vertex Type, Small-Domain Constrained Search. Why the shapes repeat →