Competition · AMC preparation · step 4 of 4
AMC 8 · 2019 · #14
Grade 4 number-theorylogicPick an answer.
AMC 8 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Find the 10-day weekday shift
A week is 7 days, so only the leftover past a full week moves the weekday. As 10 ÷ 7 leaves 3, each jump shifts the weekday forward 3 days.
Grade 4 long division with remainders tells us the weekday slides forward by 3 each time.
4.NBT.B.6Look For A PatternList every redemption weekday
Label the first weekday D, then add 3 each jump, subtracting 7 whenever you pass a full week, giving offsets D, D+3, D+6, D+2, D+5, D+1.
Generating a number pattern by a fixed rule (+3, then wrap around at 7) is the Grade 4 "shape and number pattern" standard.
4.OA.C.5Make A Systematic ListFind the missing offset
Sorted, the six offsets are D, D+1, D+2, D+3, D+5, D+6 — six of the seven possible offsets {0..6}; the one 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.
3.OA.D.9Look For A PatternApply the no-Sunday rule
The skipped weekday must be Sunday, so D+4 is Sunday; counting 4 weekdays back from Sunday (Sat, Fri, Thu, Wed) gives D = Wednesday.
Counting backward 4 weekdays from Sunday is a Grade 4 pattern-following step — no algebra needed.
Since the six redemption days land on six different weekdays and none of them is Sunday, the single weekday the schedule skips must be Sunday, and that pins down the weekday of the first redemption.
▸ Why?
The six redemptions fall on six different weekdays, so out of the seven weekdays exactly one is left untouched.
▸ Why?
Sort the seven weekdays into the ones that get a redemption and the one that does not; six different weekdays are used, and seven minus six leaves exactly one unused.
▸ Why?
Each redemption is 3 weekdays past the one before it, and stepping forward by 3 around a 7-day week reaches six different weekdays before anything repeats.
▸ Why?
Ten days is one whole week plus 3 leftover days, and a whole week ends on the same weekday it started, so only the 3 leftover days move the weekday.
▸ Why?
The weekday names repeat in a 7-day cycle, so after a full 7 days you are back on the same weekday and those full weeks can be ignored.
▸ Why?
Ten separates cleanly into 7 and 3 with nothing left over, so the part that actually moves the weekday is exactly 3.
▸ Why?
If two of the first six landings were the same weekday, the 3-day jumps between them would together make an exact number of whole weeks — a multiple of 7. But the prime factors of that stack of 3-day jumps are only 3s and the count of jumps, with no 7 hiding among the 3s, so a 7 can appear only once the count of jumps is itself 7; the first six jumps never reach that, so the six landings stay all different.
▸ Why?
The skipped weekday sits a fixed number of days ahead of the first redemption, so once the skip is known to be Sunday we count back that many days to reach the start.
▸ Why?
The unused weekday is 4 days ahead of the first redemption, so the first redemption is that weekday counted back 4 days, because counting back undoes counting forward.
Rule out the other days
Testing each start, only Wednesday skips Sunday — Monday skips Friday, Tuesday Saturday, Thursday Monday, Friday Tuesday — confirming (C).
Plugging each multiple-choice candidate back into the rule is the safest AMC "check the answer" habit.
4.OA.C.5Eliminate PossibilitiesThis AMC 8 problem only needs Grade 4 division-with-remainders and pattern-counting that you already know!
- Find the 10-day weekday shift
- List every redemption weekday
- Find the missing offset
- Apply the no-Sunday rule
- Rule out the other days
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