Problem
Find the nth largest card number
Four cards show 3, 5, 6 and 4. Use all four cards once each to build a four-digit number. Keep only the numbers between 4500 and 5450. Find every such number.
Reasoning · Grade 4-1 Making Many Kinds of Numbers
Your answer
Watch out 5436 and 5463 look alike, but 5463 is above 5450 and drops out.
How to solve
Strategy Make a Systematic List — There are 24 rearrangements of four cards, which is too many to write out safely by hand. Instead I decide the thousands digit first, because that digit alone already fixes which numbers can possibly fall in the range, and most choices die immediately. That leaves two small subproblems — numbers starting with 4 and numbers starting with 5 — and inside each one the hundreds digit narrows things again, so only a handful of numbers ever need to be written out and checked.
1STEP 1
Decide the thousands digit first
Leading 3 or 6 falls outside, leaving only 4 and 5.
3654 ＜ 4500 and 6345 ＞ 5450
A thousand is worth ten times a hundred, so no rearrangement of the last three digits can ever make up a whole thousand of difference — the leading digit settles the comparison by itself.
4.NBT.A.1Eliminate PossibilitiesThe thousands digit settles the comparison by itself, because no rearranging of the last three digits can make up a whole thousand.
Why?
A thousand is worth ten times a hundred, so the three places below the thousands can never climb past 999 between them.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
The number is what all four place amounts come to together, so a difference of one in the thousands outweighs everything beneath it.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
2STEP 2
Case 1: the number starts with 4
Leading 4 gives 4536, 4563, 4635, 4653.
4536, 4563, 4635, 4653
Once the first two digits are chosen, only two cards are left, and two cards make exactly two orders — a list short enough to write out with no chance of missing one.
4.NBT.A.2Identify Subproblems3STEP 3
Case 2: the number starts with 5
Leading 5 must stay under 5450: 5346, 5364, 5436.
5436 ＜ 5450 ＜ 5463
Comparing 5436 with 5450 means comparing digit by digit from the left: the thousands and hundreds digits match, so the tens digit decides — 3 is less than 5, while 6 is greater than 5.
4.NBT.A.2Identify Subproblems4STEP 4
Collect the whole list in order
All together that is 7 numbers.
4 + 3 = 7
Sorting the finished list from smallest to largest is a last safety check: a duplicate, or a number outside the range, would stand out immediately in an ordered list.
4.NBT.A.2Make A Systematic ListAnswer
4536, 4563, 4635, 4653, 5346, 5364, 5436
4 + 3 = 7
Every number listed uses each of the cards 3, 4, 5 and 6 exactly once, and every one lies between 4500 and 5450: the smallest is 4536 and the largest is 5436. The count is believable too — the four cards make 24 arrangements altogether, spanning 3456 up to 6543, and the allowed window is only about 950 wide inside that span of nearly 3100, so a handful of numbers is exactly what to expect. The near-misses 4365 (too small) and 5463 (too big) sit just outside the boundaries, which shows the boundaries really were tested.
Takeaway
Decide the biggest place first — once you know the number starts with 4 or 5, only a few arrangements are left to check!
- Decide the thousands digit first
- Case 1: the number starts with 4
- Case 2: the number starts with 5
- Collect the whole list in order