AMC 8 · 2019 · #14
Easy mode Grade 4Problem
Imagine a calendar in front of you. Isabella has 6 coupons for free ice cream cones at Pete's Sweet Treats.
She plans to use one coupon every 10 days, until all 6 are used. So she circles 6 dates on her calendar: the first day, then 10 days later, then 10 days after that, and so on.
Pete's is closed on Sundays. When Isabella looks at her 6 circled dates, none of them fall on a Sunday.
What day of the week is the first circled date?
Pick an answer.
AMC 8 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Isabella has $6$ free-ice-cream coupons and decides to use one every $10$ days until they are gone. Pete's is closed on Sundays, and when she circles all $6$ redemption dates on the calendar, not a single one lands on a Sunday. On which day of the week does she redeem the first coupon?
Givens: Isabella has $6$ coupons total; She redeems one coupon every $10$ days (so $6$ redemptions span days $0, 10, 20, 30, 40, 50$ from the first); Pete's is closed on Sundays; None of the $6$ circled redemption dates falls on a Sunday; Answer choices: (A) Monday, (B) Tuesday, (C) Wednesday, (D) Thursday, (E) Friday
Unknowns: The day of the week on which Isabella redeems her FIRST coupon
Understand
Restated: Isabella has $6$ free-ice-cream coupons and decides to use one every $10$ days until they are gone. Pete's is closed on Sundays, and when she circles all $6$ redemption dates on the calendar, not a single one lands on a Sunday. On which day of the week does she redeem the first coupon?
Givens: Isabella has $6$ coupons total; She redeems one coupon every $10$ days (so $6$ redemptions span days $0, 10, 20, 30, 40, 50$ from the first); Pete's is closed on Sundays; None of the $6$ circled redemption dates falls on a Sunday; Answer choices: (A) Monday, (B) Tuesday, (C) Wednesday, (D) Thursday, (E) Friday
Plan
Primary tool: #5 Look for a Pattern
Secondary: #2 Make a Systematic List, #3 Eliminate Possibilities
Days of the week form a $7$-day cycle, so jumping $10$ days really means jumping $10 - 7 = 3$ days forward each time. Tool #5 (Look for a Pattern) lets us discover that hidden "+3 each jump" rule. Tool #2 (Systematic List) then lays out all $6$ redemption weekdays in order so we can SEE which weekdays are covered and which one is skipped. Finally, since the problem is multiple-choice and the "no Sunday" rule eliminates four of the five candidates, Tool #3 (Eliminate Possibilities) confirms the answer quickly.
Execute — Answer: C
4.NBT.B.6 Step 1 - Find how much the weekday shifts every $10$ days.
- Because the week itself is $7$ days long, full weeks don't change the weekday — only the LEFTOVER days do.
- Divide $10$ by $7$ to find that leftover.
💡 Grade 4 long division with remainders tells us the weekday slides forward by $3$ each time.
4.OA.C.5 Step 2 - Use Tool #2 to list the weekday of every redemption in order, calling the first weekday $D$.
- After each jump, add $3$ days; whenever the running total goes past a full week ($7$ days), subtract $7$ to come back into the same week.
💡 Generating a number pattern by a fixed rule ($+3$, then wrap around at $7$) is the Grade 4 "shape and number pattern" standard.
3.OA.D.9 Step 3 - Collect those $6$ offsets in order from smallest to largest.
- They are $\{D+0,\; D+1,\; D+2,\; D+3,\; D+5,\; D+6\}$ — six of the seven possible weekday offsets $\{0,1,2,3,4,5,6\}$.
- The ONE offset that is missing is $D+4$.
💡 Spotting that $\{0,1,2,3,5,6\}$ is the full set $\{0,1,2,3,4,5,6\}$ minus $4$ is a Grade 3 arithmetic-pattern observation.
4.OA.C.5 Step 4 - Apply the "no Sundays" rule.
- The $6$ redemption days hit $6$ of the $7$ weekdays and skip exactly one, and the problem says the skipped weekday must be Sunday.
- So $D + 4$ lands on Sunday, which means $D$ itself is $4$ days BEFORE Sunday — count backward from Sunday: Saturday, Friday, Thursday, Wednesday.
- So $D = $ Wednesday.
💡 Counting backward $4$ weekdays from Sunday is a Grade 4 pattern-following step — no algebra needed.
4.OA.C.5 Step 5 - Use Tool #3 to eliminate the other choices.
- (A) Monday $\to$ skipped day is Friday, not Sunday.
- (B) Tuesday $\to$ skipped day is Saturday.
- (D) Thursday $\to$ skipped day is Monday.
- (E) Friday $\to$ skipped day is Tuesday.
- Only (C) Wednesday makes the skipped day Sunday, matching the puzzle's rule.
💡 Plugging each multiple-choice candidate back into the rule is the safest AMC "check the answer" habit.
4.NBT.B.6 Find how much the weekday shifts every $10$ days. Because the week itself is $7$ 4.OA.C.5 Use Tool #2 to list the weekday of every redemption in order, calling the first 3.OA.D.9 Collect those $6$ offsets in order from smallest to largest. They are ${D+0,\; 4.OA.C.5 Apply the "no Sundays" rule. The $6$ redemption days hit $6$ of the $7$ weekdays 4.OA.C.5 Use Tool #3 to eliminate the other choices. (A) Monday $\to$ skipped day is Frid Review
Reasonableness: Verify by hand starting from Wednesday: Wed $\to$ Sat $\to$ Tue $\to$ Fri $\to$ Mon $\to$ Thu. That's $\{$Wed, Sat, Tue, Fri, Mon, Thu$\}$ — exactly $6$ different weekdays, and the only weekday missing is Sunday. Perfect match with the puzzle's "no Sunday" rule, so Wednesday is correct.
Alternative: Tool #6 (Guess and Check) on each choice: for each of the five starting weekdays, list the six $+3$-stepped weekdays and check whether Sunday appears. Only Wednesday produces a Sunday-free schedule, so Wednesday wins. This is even more direct than the pattern argument and is a great backup whenever the "missing-day" trick feels slippery.
CCSS standards used (min grade 4)
3.OA.D.9Identify arithmetic patterns and explain using properties of operations (Noticing that the six computed weekday offsets $\{0,1,2,3,5,6\}$ are missing exactly one value ($4$) from the full set $\{0,1,2,3,4,5,6\}$.)4.NBT.B.6Find whole-number quotients and remainders with up to four-digit dividends (Dividing $10 \div 7$ to get a remainder of $3$, which is the weekday shift per redemption.)4.OA.C.5Generate a number or shape pattern following a given rule (Building the list of six redemption weekdays by repeatedly adding $3$ and wrapping past $7$, and counting backward $4$ days from Sunday to find the starting weekday.)
⭐ This AMC 8 problem only needs Grade 4 division-with-remainders and pattern-counting that you already know!
⭐ This AMC 8 problem only needs Grade 4 division-with-remainders and pattern-counting that you already know!
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