AMC 10 · 2023 · #9

Grade 4 arithmetic

Archetype Small-Domain Constrained Search

digit-constraintsparitysystematic-enumerationplace-value caseworksystematic-enumerationidentify-subproblems ↑ Prerequisites: digit-constraintsparity

📏 Long solution 💡 3 insights

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Problem

A digital display shows the current date as an $8$ -digit integer consisting of a $4$ -digit year, followed by a $2$ -digit month, followed by a $2$ -digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in $2023$ will each digit appear an even number of times in the 8-digital display for that date?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: In the $8$-digit display $\text{YYYYMMDD}$, the year part is fixed as $2023$. Count the dates within $2023$ for which every digit from $0$ to $9$ appears an even number of times across the entire $8$-digit string.

Givens: Year digits: $2, 0, 2, 3$ — the digit $2$ appears twice, $0$ once, $3$ once, all others zero times; Month $\text{MM}$ is between $01$ and $12$; Day $\text{DD}$ is a valid day for that month in $2023$ (not a leap year, so February has $28$ days); Answer choices: (A) $5$, (B) $6$, (C) $7$, (D) $8$, (E) $9$

Unknowns: The number of valid dates $\text{MMDD}$ such that the digit count of $\text{YYYYMMDD}$ is even for every digit

Understand

Plan

Primary tool: #15 Organize Information in More Ways

Secondary: #2 Make a Systematic List, #7 Identify Subproblems

The first move is Tool #15 (Reorganize) — drop the calendar story and re-frame the question as a digit-parity bookkeeping problem on $\text{MMDD}$. The year $2023$ already has odd counts only at digits $0$ and $3$, so the four digits of $\text{MMDD}$ must include an odd count of $0$'s, an odd count of $3$'s, and an even count of every other digit. With four slots that pins the digit multiset down to a tiny set of cases — perfect for Tool #2 (Systematic List). Tool #7 (Identify Subproblems) handles the calendar filter: for each candidate digit multiset, list the rearrangements that form a valid $\text{MM}/\text{DD}$.

Execute — Answer: E

#15 Organize Information in More Ways 2.OA.C.3 Step 1

Count the digits of the year $2023$.
The multiset is $\{2, 0, 2, 3\}$, so the digit $2$ appears twice (even), $0$ once (odd), $3$ once (odd), and every other digit zero times (even).

$$\text{year parity}: 0 \to \text{odd}, \; 3 \to \text{odd}, \; \text{else} \to \text{even}$$

💡 Sort the year's digits and count each — Grade 2 odd/even classification.

#15 Organize Information in More Ways 2.OA.C.3 Step 2

For the full $8$-digit string to have every digit appearing an even number of times, the four $\text{MMDD}$ digits must flip the year's parities.
So the $\text{MMDD}$ digit multiset must contain an odd number of $0$'s, an odd number of $3$'s, and an even number of every other digit.

$$\#\{0\}_{\text{MMDD}} \in \{1, 3\}, \; \#\{3\}_{\text{MMDD}} \in \{1, 3\}, \; \#\{d\}_{\text{MMDD}} \in \{0, 2, 4\} \text{ for } d \notin \{0, 3\}$$

💡 Odd + odd = even; the four extra digits must "fix" each odd column. Parity arithmetic, Grade 2.

#2 Make a Systematic List 4.OA.C.5 Step 3

List the digit multisets compatible with the parity rule on exactly four slots.
The odd counts of $0$'s and $3$'s alone use $1+1 = 2$, $1+3 = 4$, or $3+1 = 4$ slots; any extra slots must come in even pairs of some other digit $d$.
The valid multisets are $\{0, 3, d, d\}$ for some digit $d$, $\{0, 3, 3, 3\}$, and $\{0, 0, 0, 3\}$.

$$\text{cases: } \{0, 3, d, d\}, \; \{0, 3, 3, 3\}, \; \{0, 0, 0, 3\}$$

💡 List the possibilities by case — Grade 4 "generate by a rule" enumeration.

#7 Identify Subproblems 3.OA.D.8 Step 4

Subproblem A — multiset $\{0, 3, 3, 3\}$ and $\{0, 0, 0, 3\}$.
Any arrangement into $\text{MM} \text{DD}$ either gives $\text{MM} \ge 13$ or $\text{DD}$ with a leading $3$ ($\ge 30$ in a month with three $3$'s or $0$'s), or has $\text{MM} = 00$ or $\text{DD} = 00$.
Check the candidates: from $\{0,3,3,3\}$ the only $\text{MM}$ values $\le 12$ are $03$ and $30$ — but $30 > 12$ — leaving $\text{MM}=03$ and $\text{DD}=33$, invalid.
From $\{0,0,0,3\}$ valid $\text{MM}$ would be $00$ or $03$ or $30$; only $03$ qualifies, giving $\text{DD}=00$, invalid.
Zero dates from these two multisets.

$$\{0,3,3,3\}: 0, \quad \{0,0,0,3\}: 0$$

💡 Real calendar rules — months $01$–$12$, no day $00$ or $> 31$ — kill these multisets. Grade 3 multi-step reasoning.

#7 Identify Subproblems 4.OA.B.4 Step 5

Subproblem B — multiset $\{0, 3, d, d\}$ with $d \ne 0, 3$.
For $\text{MM}$ to be valid ($01$–$12$), the digit $d$ must let $\text{MM}$ stay $\le 12$.
Check $d$ by case: $d = 1$ works (gives months $01$, $10$, $11$, $03$); $d = 2$ works ($\text{MM} = 02, 03, 12$ possible); $d \ge 4$ fails because $\text{MM}$ would need to use two large digits and still be $\le 12$, but the two $d$'s can only form $\text{MM}=dd \ge 44$ unless one of $\{0, 3\}$ takes the lead — and then $\text{DD}$ would need digits $\ge 30$ that exceed any month's day count.

$$\text{candidate } d: \; 1, \; 2$$

💡 Filter $d$ by the constraint that some arrangement gives $\text{MM} \le 12$ — Grade 4 factor-style case checking.

#2 Make a Systematic List 4.OA.A.3 Step 6

Sub-list $d = 1$: digit multiset $\{0, 1, 1, 3\}$.
Pick $\text{MM}$ from valid month-strings using two of the four digits, then $\text{DD}$ from the remaining two.
Valid $(\text{MM}, \text{DD})$ pairs that also form a real calendar day: $(01, 13), (01, 31), (03, 11), (10, 13), (10, 31), (11, 03), (11, 30)$.
Seven dates.

$$d=1: \; \{0113, 0131, 0311, 1013, 1031, 1103, 1130\} \Rightarrow 7$$

💡 Systematic listing of arrangements that pass the month/day filter — Grade 4 multi-step word problem.

#2 Make a Systematic List 4.OA.A.3 Step 7

Sub-list $d = 2$: digit multiset $\{0, 2, 2, 3\}$.
Try every valid month: $\text{MM} = 02$ leaves $\{2, 3\}$ for $\text{DD}$ giving $23$ or $32$ — only $23$ is valid (February has $28$ days in $2023$).
$\text{MM} = 03$ leaves $\{2, 2\}$ giving $\text{DD} = 22$, valid.
$\text{MM} = 12$ leaves $\{0, 3\}$ giving $03$ or $30$ — but the multiset requires two $2$'s, and using $12$ as month consumes only one $2$, leaving one $2$ unused; actually $\text{MM} = 12$ doesn't fit the multiset $\{0,2,2,3\}$ at all since it would use digits $1, 2$ and we have no $1$.
Final count: $(02, 23)$ and $(03, 22)$.
Two dates.

$$d=2: \; \{0223, 0322\} \Rightarrow 2$$

💡 Same enumeration discipline as $d=1$, with the calendar killing the $32$ candidate. Grade 4 multi-step.

#7 Identify Subproblems 2.NBT.B.5 Step 8

Combine.
Multiset $\{0,1,1,3\}$ gives $7$ dates, $\{0,2,2,3\}$ gives $2$ dates, and the other multisets give $0$.
Total $7 + 2 = 9$, matching choice (E).

$$7 + 2 = 9 \;\Rightarrow\; \textbf{(E)}$$

💡 Two case totals to add — Grade 2 fluency within $100$.

[1] #15 2.OA.C.3 Count the digits of the year $2023$. The multiset is $\{2, 0, 2, 3\}$, so the di

[2] #15 2.OA.C.3 For the full $8$-digit string to have every digit appearing an even number of ti

[3] #2 4.OA.C.5 List the digit multisets compatible with the parity rule on exactly four slots.

[4] #7 3.OA.D.8 Subproblem A — multiset $\{0, 3, 3, 3\}$ and $\{0, 0, 0, 3\}$. Any arrangement i

[5] #7 4.OA.B.4 Subproblem B — multiset $\{0, 3, d, d\}$ with $d \ne 0, 3$. For $\text{MM}$ to b

[6] #2 4.OA.A.3 Sub-list $d = 1$: digit multiset $\{0, 1, 1, 3\}$. Pick $\text{MM}$ from valid m

[7] #2 4.OA.A.3 Sub-list $d = 2$: digit multiset $\{0, 2, 2, 3\}$. Try every valid month: $\text

[8] #7 2.NBT.B.5 Combine. Multiset $\{0,1,1,3\}$ gives $7$ dates, $\{0,2,2,3\}$ gives $2$ dates,

Review

Reasonableness: Spot-check each of the $9$ dates by concatenating with $2023$ and tallying digits: e.g. $20230113 \to$ digits $\{2,0,2,3,0,1,1,3\}$: $0$ twice, $1$ twice, $2$ twice, $3$ twice — all even ✓. Similarly $20230131$: $\{2,0,2,3,0,1,3,1\}$ — $0$ twice, $1$ twice, $2$ twice, $3$ twice ✓. $20230223 \to \{2,0,2,3,0,2,2,3\}$: $0$ twice, $2$ four times, $3$ twice ✓. The same parity verification works on the rest. Magnitude check: with $9$ qualifying dates out of $365$ days in $2023$, the rate is about $2.5\%$ — a believable order of magnitude for a fairly restrictive parity constraint.

Alternative: Tool #6 (Guess & Check) — direct calendar scan: walk through each of the $12$ months, and for each month enumerate the valid days, concatenate with $2023$ and tally digit counts on the fly. Slower (you check $\sim 365$ candidates instead of $\sim 20$ multiset arrangements), but it requires no parity reasoning — only patience and a list. The answer comes out the same: $9$.

CCSS standards used (min grade 4)

2.OA.C.3 Determine whether a group of objects has an odd or even number (Tallying parities of each digit in $2023$ ($0$: odd, $3$: odd, $2$: even, rest: even) and propagating that parity bookkeeping to $\text{MMDD}$.)
4.OA.C.5 Generate a number or shape pattern following a given rule (Enumerating the digit-multisets on $4$ slots compatible with the parity rule — $\{0,3,d,d\}$, $\{0,3,3,3\}$, $\{0,0,0,3\}$.)
4.OA.B.4 Find all factor pairs / consider all possibilities for a whole number (Filtering the family $\{0, 3, d, d\}$ on the value of $d$ so that some arrangement yields a valid month $\text{MM} \le 12$.)
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers (Inside each multiset, building all valid $(\text{MM}, \text{DD})$ assignments while honoring calendar constraints (month $\le 12$, day $\le$ days-in-month).)
3.OA.D.8 Solve two-step word problems using the four operations (Showing that multisets $\{0,3,3,3\}$ and $\{0,0,0,3\}$ produce no valid calendar date by checking each possible $\text{MM}/\text{DD}$ split.)
2.NBT.B.5 Fluently add and subtract within 100 (Adding the case totals $7 + 2 = 9$ at the end.)

⭐ This AMC 10 problem only needs Grade 4 pattern enumeration you already know — make $2023$'s odd-count digits ($0$ and $3$) come out even by carefully choosing the four $\text{MMDD}$ digits, then list the calendar dates that fit.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Review

CCSS standards used (min grade 4)

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