AMC 10 · 2023 · #20
Grade 9 patternnumber-theory
Pick an answer.
Row 2023 has 2023 separate entries, and not one of them is wanted — only their total is. Tool #9 (Solve an Easier Related Problem) is the opening move: replace the question "what are the 2023 numbers?" with "what is their sum?", which is one number instead of 2023. Tool #4 (Introduce a Variable) then does the real work twice over. First it names that one number, S_n, so a row-to-row rule can even be written. Second, and this is the crux, it renames S_n as T_n = S_n + n, which turns a messy recurrence with a drifting +n-2 into plain doubling. Tool #16 (Change Focus) supplies the recurrence itself: instead of adding up row n entry by entry, count how many times each entry of row n-1 gets used by row n. Tool #5 (Look for a Pattern) closes the problem, since the units digits of powers of 2 repeat in a cycle of four. Tool #15 (Organize Information in More Ways) keeps a running row-sum table that every later claim can be tested against.
Tabulate the row sums
Tabulate the first few sums.
Generating one extra row by hand converts a written rule into something concrete enough to argue about, and the sums table becomes a permanent answer key for later formulas.
4.OA.C.5Organize Information In More WaysChase the sum, not the entries
Chase the sum, not the entries.
A sequence indexed by row number is a far smaller object than the triangle itself, and it is all the question ever asks about.
9.F-IF.A.3Solve An Easier Related ProblemHow often each parent is used
Interior parents are used twice.
Summing a row by counting how often each parent is reused, rather than by adding entries one at a time, is what makes the unknown entries cancel out of the bookkeeping.
9.A-SSE.A.2Change Focus Count The ComplementShift the sequence
A small shift simplifies the rule.
A recurrence that doubles and then adds a changing amount is a doubling recurrence in disguise, once the sequence is measured from the right starting line.
9.F-BF.A.1Introduce A VariableRead the doubling rule
Doubling gives a power of two.
Repeated doubling from a known starting value is exactly what an exponential expression records, so one anchor plus the doubling rule pins down every term at once.
9.F-LE.A.2Look For A PatternUse the units-digit cycle
Powers of two cycle through four units digits.
Each doubling depends on the previous units digit alone, so the last digits are trapped in a short loop no matter how large the exponent grows.
Each doubling depends only on the previous last digit, so the last digits are trapped in a short loop.
▸ Why?
Once a last digit repeats, everything that followed it the first time follows it again in the same order.
▸ Why?
Carrying only ever moves left, so the last digit is sealed off from every higher place.
Subtract the last digits
Subtracting gives 5.
Last digits add and subtract among themselves, so a six-hundred-digit subtraction collapses into 8 - 3.
7.NS.A.1Change Focus Count The ComplementCheck on real rows
Confirm the formula on real rows.
A formula that reproduces every row anyone can actually count is a formula worth trusting on a row nobody can.
4.OA.C.5Organize Information In More WaysWhen a rule builds each row from the one above, stop tracking the entries and track only the row sum — here shifting it to S_n + n makes it double every row, so S_n = 2ⁿ - n and the last digit of 2²⁰²³ - 2023 is 8 - 3 = 5.
- Tabulate the rows and their sums
- Chase the sum, not the entries
- Count how often each parent is used
- Shift the sequence to kill the drift
- Doubling gives a power of two
- Units digits of powers of two cycle by four
- Subtract the last digits
- Confirm the formula on real rows