Problem
Ascending and descending digit numbers
Five numbers are given: 34, 16, 248, 359 and 2457. Find the property all five share. Then look only at four-digit numbers with that property. Write every one of them bigger than 5000.
Reasoning · Grade 4-1 Making Many Kinds of Numbers
Your answer
Watch out Any digit 5 to 9 seems fine, but 7, 8 and 9 run out of larger digits.
How to solve
Strategy Look for a Pattern — Part (1) is a hidden-rule question, so I compare the digits inside each example until one description fits all five numbers at once. Once the rule is known, part (2) becomes a search. There are five thousand four-digit numbers above 5000, far too many to check one at a time, so I first use the rule to throw out whole blocks of thousands digits, and only then list what survives — in a fixed order, so that nothing is missed and nothing is written twice.
1STEP 1
Compare the digits inside each example
In all five the digits increase left to right.
3 ＜ 4, 1 ＜ 6, 2 ＜ 4 ＜ 8, 3 ＜ 5 ＜ 9, 2 ＜ 4 ＜ 5 ＜ 7
Once you know a numeral is just a string of digits, comparing neighbouring digits is something you can do by eye — no calculation is needed at all.
4.NBT.A.2Look For A Pattern2STEP 2
State the rule and notice what it forbids
So no digit repeats and 0 can never appear.
Turning a pattern into a rule about what is forbidden is what makes the search in part (2) short — the rule does the work of testing thousands of numbers.
4.NBT.A.2Look For A Pattern3STEP 3
See which thousands digits are even possible
Bigger than 5000 needs a thousands digit of 5 or more.
Counting how many digits are still available above the first one is just reading off part of the 0-to-9 number line.
4.NBT.A.2Eliminate Possibilities4STEP 4
Throw out the thousands digits that cannot work
7, 8 and 9 lack enough larger digits, so they drop out.
7→{8,9}: 2 ＜ 3, 8→{9}: 1 ＜ 3, 9→{ }: 0 ＜ 3
Comparing how many digits are needed with how many are left settles three whole blocks of a thousand numbers each, without writing a single one of them down.
4.NBT.A.2Eliminate PossibilitiesComparing how many digits are still needed with how many are still available rules out whole blocks of a thousand numbers at once.
Why?
Each digit after the first must be strictly larger, so the digits climb and there must be enough room above the leading digit to fit them all.
🧱Transitivity of orderIf the first is smaller than the second and the second smaller than the third, the first is smaller than the third, so pairwise comparisons chain into one full order.
Why?
Any leading digit that leaves too few larger digits above it is thrown out immediately, without writing a single number down.
🧱Rule out the rest and the last one is forcedIf every possibility but one has been ruled out, the survivor must be the answer — no further checking is needed.
5STEP 5
List every number starting with 5
Starting with 5 gives 5678, 5679, 5689, 5789.
5678, 5679, 5689, 5789
Going through the choices in a fixed order — leave out the biggest, then the next, and so on — is what guarantees the list is complete.
4.OA.C.5Make A Systematic List6STEP 6
List every number starting with 6
Starting with 6 gives only 6789.
6789
When the number of things you need is the same as the number you have, there is only one way to do it — no listing required.
4.OA.C.5Make A Systematic List7STEP 7
Collect the two cases
Together that is 5 numbers.
4 + 1 = 5
Adding the counts from the separate cases is safe precisely because the cases were split by the first digit, so no number can turn up in two of them.
4.OA.C.5Make A Systematic ListAnswer
5678, 5679, 5689, 5789, 6789
4 + 1 = 5
Each listed number really is greater than 5000, and each really does have increasing digits: 5 ＜ 6 ＜ 7 ＜ 8, 5 ＜ 6 ＜ 7 ＜ 9, 5 ＜ 6 ＜ 8 ＜ 9, 5 ＜ 7 ＜ 8 ＜ 9, 6 ＜ 7 ＜ 8 ＜ 9. There cannot be many such numbers, since squeezing four different increasing digits out of only 5, 6, 7, 8, 9 leaves very little room, so a total of five is the right size. It also matches a count of digit sets: sets of four digits from {5,6,7,8,9} that begin with 5 or 6 number 4 + 1 = 5, and each set can be written in increasing order in only one way.
Takeaway
Once you name the hidden rule, it does the searching for you — "the digits must keep growing" wipes out thousands of numbers and leaves just five!
- Compare the digits inside each example
- State the rule and notice what it forbids
- See which thousands digits are even possible
- Throw out the thousands digits that cannot work
- List every number starting with 5
- List every number starting with 6
- Collect the two cases