Hint
The largest possible difference is when 1 is next to 8, a difference of 7. Since we have 7 differences to find, and the largest possible difference is 7, all of the possible differences must exist. Start by looking where can the 8 go.
Answer
Reasoning
The largest possible difference is when 1 is next to 8, a difference of 7. Since we have 7 differences to find, and the largest possible difference is 7, all of the possible differences must exist: 1, 2, 3, 4, 5, 6, 7, and let's call these D1, …, D7.
D7 can only happen when: 1 is next to 8 = D7
D6 can happen when:
1 is next to 7, but these are given numbers that are not next to each other. 2 is next to 8 = D6
Where can 8 go? If we put 8 above 1, we cannot also satisfy D6 (2 is next to 8).
Therefore, we have two possibilities:
(a) 8 to the left of 1
(b) 8 to the right of 1 (a) 8 to the left of 1
By D6, 2 would be below 8, and this would give us D1, D6, D7. What can we place to the right of 1?
Not 3, because the difference between 1 and 3, and the difference between 3 and 5, are both D2.
Not 4, because the difference between 4 and 5 is D1, which we would already have.
Not 6, because the difference between 5 and 6 is D1, which we would already have.
There are no possible numbers we can place to the right of 1, so 8 can't go to the left of 1. (b) 8 to the right of 1
By D6, 2 would be above 8, and this would give us D3, D6, D7.
4 can't go next to 1, otherwise we'd create another D3. Therefore, 4 goes in the bottom left corner.
We are now left with 3 and 6.
If 6 went above 4, and 3 above 1, these would both be D2.
Therefore, 3 goes above 4, 6 goes above 1.
The final answer is:
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Puzzle 19
If cat and a half can catch a mouse and a half in a day and a half how many mice can 3 cats catch in 3 days?
Alex and Billie were rowing their canoe along the River Trent.
In the morning, they managed to row upstream at an average speed of 2 miles per hour.
They then stopped for a spot of lunch and a nice rest.
In the afternoon, the pace was a little easier as they were now rowing downstream back to their starting point, and managed an average speed of 4 miles an hour.
The morning trip took them 3 hours longer than the afternoon.
Hint
You can either fix the distance rowed and see what numbers work, or you can fix the number of hours.
Answer
12 miles.
In the morning, rowing at 2 miles per hour, they rowed for 6 hours. In the afternoon, rowing at 4 miles per hour, they rowed for 3 hours.
There are a number of ways of working this out, and here are two of them:
Method 1
If we assume the distance the rowed upstream to be D miles, we know the morning took (D ÷ 2) hours, and the afternoon took (D ÷ 4) hours, with a difference of 3 hours. So:
D D
- − - = 3
2 4
Multiplying throughout by 4 gives:
2D − D = 12
So:
D = 12 miles
They rowed 12 miles upstream.
Method 2
If we assume they rowed for H hours upstream, we know they travelled H x 2 miles in the morning. In the afternoon, they rowed for (H − 3) hours, and travelled (H − 3) x 4 miles. We know these distances are the same, so:
2H = (H − 3) x 4
Giving:
2H = 4H − 12
Rearranging gives:
12 = 2H
So:
H = 6 hours
They rowed for 6 hours upstream at 2 miles per hour, which is a total of 12 miles.
