The coach likes logic problems where one of the participants is blindfolded but, nevertheless, can still prevail over the other participants.
Sevens and Elevens
Participants are told there are nine slips of paper. Five of those slips have the number 7 on them, while the other four have 11 on them. Alice, Bob, Carl, Dave and Edward have one of the slips on each of their hats. The other four slips remain hidden. Edward is blindfolded, but the others can see all numbers except their own. They are asked in turn to identify their number. All participants hear the responses of any that go before them.
Alice: “I don’t know my number.”
Bob: “I don’t know my number.”
Carl: “I don’t know mine.”
Dave: “I don’t know mine.”
Edward: “I know my number.”
What number is on Edward?
Each of team members Alice, Bob, and Carl wears a hat with a single digit from 0 to 9 on it. In an unknown order, they are the digits of a square having three digits. Alice and Carl can see the digits on the other two hats but not their own. Bob is blindfolded. After being asked to deduce their digits, there is quite a long silence after which Bob announces, “I know my digit.”
What digit is on Bob and what are the possibilities for digits on Alice and Carl?
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Solutions to week 135
The solution to Batting Order is 385274961 and 941638527. In Swimming Workout, Randall swam 2 miles in total at a speed of 0.8 miles an hour in the lake.
Batting Order answer explained:
There are 24 ways to place the even-numbered jerseys in the even-numbered batting positions. Examination of these produces only two ways to meet the conditions of the problem: 385274961 and 941638527.
Swimming Workout answer explained:
Suppose the distance Randall swam upstream was s. Then the time swimming the stream portions was t = s/1.2 + s/0.6 and the average speed on these segments was 2s/t = 2 × 1.2 × 0.6/(1.2 + 0.6) = 0.8 mph. If the distance across the lake from the campground to the mouth of the stream is d, then Sophie can note that the total swimming time is 2½ = 2d/v + 2s/0.8, where v is Randall’s normal swimming speed in the lake. The total distance Randall swam is 2d + 2s and the only way Sophie could have computed this is if she knew his normal swimming speed in the lake was 0.8 mph. We now know his swim speed on the lake must be 0.8 mph so that the total distance Randall swam was D = (2½) hours × 0.8 mph = 2 miles.
Week 135: Swimming Workout & Batting Order, solutions to Maximization & Multiplication Table
Week 134: Maximization & Multiplication Table, solutions to Maximizing Links & Seven Points
Week 133: Maximizing Links & Seven Points, solutions to Time of Day & The Missing Element
Week 132: Time of Day & The Missing Element, solutions to Knight Trap & My Number is 136
Week 131: Knight Trap & My Number is 136, solutions to The Size of Humanity & Fish Pond
Links to all of the puzzles and solutions are on the Complete Varsity Math page.
Come back next week for answers and more puzzles.