Digits are written as digital-clock numerals, so a number's digits side by side make one picture. Find the second smallest 3-digit number whose picture is line-symmetric, the second largest 4-digit one that is point-symmetric, and how many of those lie between 5000 and 8000.
Your answer
Watch outJudging digit by digit misses that a half turn also reverses their order.
How to solve
StrategyMake a Systematic List — Before touching the numbers I sort the ten numerals into three small groups: the ones that fold onto themselves, the ones a half turn leaves alone, and the special pair 6 and 9 that swap. That short table does almost all the work. After that each question is a small, finite search: for (1) I want the smallest numbers built from a short list of allowed digits, for (2) I want the largest, and for (3) the thousands digit has only 3 possible values, so I can knock out the impossible one and then list what is left in order.
1STEP 1
Sort the ten numerals by how they fold
Folding leaves 0, 1, 3, 8 unchanged.
fold onto themselves: 0, 1, 3, 8
You can test this with a mirror laid flat under the numeral: 3 and 8 look untouched, and 1 is only an upright stroke so nothing can happen to it, while 2 and 5 clearly swap places.
4.G.A.3Visualize Spatial Relationships
2STEP 2
Sort the ten numerals by how they turn
A half turn leaves 0, 1, 2, 5, 8 unchanged.
unchanged by a half turn: 0,1,2,5,8; 6 ⇔ 9
This is the digital-clock fact everyone discovers by accident when they read a clock upside down; here it is written down carefully so it can be used as a rule.
8.G.A.1Visualize Spatial Relationships
3STEP 3
Question 1: which fold line can a whole number use?
For (1) every digit must come from those four.
horizontal fold: digits from {0,1,3,8}; vertical fold: palindromes from {0,8}
The direction of the fold decides whether the digits stay in order or get reversed, and noticing that is what keeps the search short - a horizontal fold asks nothing about the order at all.
4.G.A.3Organize Information In More Ways
4STEP 4
Question 1: list the smallest ones in order
Counting up, the second smallest is 101.
100 ＜ 101 ＜ 103 ＜ 108
Comparing whole numbers place by place - hundreds first, then tens, then ones - is Grade 4 place value, and it makes 'second smallest' a matter of reading the list rather than searching.
4.NBT.A.2Make A Systematic List
5STEP 5
Question 2: what a half turn does to a whole number
For (2) the outer digits must pair up.
d₄ = turn(d₁), d₃ = turn(d₂)
The reversal is the part that catches people out: after a half turn the last digit is on the left, so the outer pair has to match up and the inner pair has to match up.
8.G.A.1Organize Information In More Ways
A half turn both reverses the order of the digits and turns each digit over, so both effects must be tracked at once.
Why?
The turn moves every digit to the opposite end of the display without stretching or bending any of them.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Why?
A numeral is a string of digits in fixed places, so swapping which digit sits in which place makes a different number.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
6STEP 6
Question 2: find the largest, then the second largest
The second largest is 9886.
9966 ＞ 9886
Making a number as large as possible always means fixing the leftmost place first and only then moving right, which is exactly the place-value order used to compare numbers.
4.NBT.A.2Make A Systematic List
7STEP 7
Question 3: rule out the impossible thousands digit
In (3) a leading 7 is impossible.
d₁ ∈ {5,6,7}; 7 eliminated
With only three candidates for the leading digit, checking them one at a time is quick and complete - and knocking out 7 immediately removes a third of the work.
4.NBT.A.2Eliminate Possibilities
8STEP 8
Question 3: count the middle pairs and add up
Counting the middle pairs gives 14.
7 + 7 + 0 = 14
Because choosing the hundreds digit settles the tens digit as well, the count is simply 'how many choices for one digit', and multiplying or adding equal groups is ordinary Grade 4 work.
4.OA.A.3Make A Systematic List
Answer
101, 9886, 14
7 + 7 = 14
Each answer can be checked directly. Turn 101 upside down on paper: it is 101 again, and folded along its middle line it is unchanged too, and the only smaller 3-digit candidate is 100, so 'second smallest' is right. Turn 9886 a half turn: the digits reverse to 6, 8, 8, 9 and each turns, giving 9, 8, 8, 6 - the same number - and the only larger one is 9966, since raising the hundreds digit from 8 to 9 is the only way up. For part 3, all 14 numbers lie strictly between 5000 and 8000 as required, each begins with 5 or 6 and ends with the matching 5 or 9, and 14 is a believable size: only 7 of the 10 numerals survive a half turn and only 2 of the 3 possible thousands digits do, so a count in the low teens is exactly what to expect.
Takeaway
Sort the ten digital numerals first - 0, 1, 3, 8 survive a fold; 0, 1, 2, 5, 8 survive a half turn and 6 and 9 trade places - and remember that a half turn also reverses the digits, so the outside pair and the inside pair each have to match.
Sort the ten numerals by how they fold
Sort the ten numerals by how they turn
Question 1: which fold line can a whole number use?
Question 1: list the smallest ones in order
Question 2: what a half turn does to a whole number
Question 2: find the largest, then the second largest
Question 3: rule out the impossible thousands digit