Puzzle 197
What number comes next in this sequence:
986888
724864
143224
==??==
Puzzle Copyright © Kevin Stone
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Hint
The answer involves multiplication (for example, 9 x 8 = 72).
Answer
468.
Reasoning
If we split each line in the following way:
98 68 88
72 48 64
14 32 24
Each new line is generated by multiplying the digits two at a time and writing the result below.
For example, 9 x 8 = 72, so 72 appears directly below 98.
And 7 x 2 = 14, so 14 appears directly below 72.
Double-Checking
9 x 8 = 72, 7 x 2 = 14, 1 x 4 = 4.
6 x 8 = 48, 4 x 8 = 32, 3 x 2 = 6.
8 x 8 = 64, 6 x 4 = 24, 2 x 4 = 8.
Puzzle 198
This BrainBashers safe has a rather unusual control panel.
In order to open the safe, you have to press every single square once only, in the correct sequence, ending on the square marked E.
Each square indicates which square to press next, for example, 2U means move up 2 squares and 3R means move right 3 squares.
Can you determine which square you must start with?
| 4R |
4D |
1R |
1D |
2D |
4L |
1D |
| 1R |
4D |
5D |
2R |
2D |
1L |
4L |
| 2R |
1D |
1L |
2R |
2R |
2D |
2U |
| 2R |
2R |
3U |
3D |
1R |
3U |
6L |
| 3U |
3R |
1D |
2U |
4L |
2D |
3L |
| 3U |
1L |
1R |
1R |
1R |
1R |
2U |
| 6U |
E |
2U |
2L |
4L |
1L |
2U |
Note: this puzzle is not interactive, and the safe buttons cannot be clicked.
Puzzle Copyright © Kevin Stone
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Hint
Start on any square, and see how far you can get.
Answer
Start from the 2U in the bottom right-hand corner.
Puzzle 199
As my autumnal birthday approaches I like to collect leaves! A little bizarre perhaps, but I enjoy it!
Starting on the first day of the month I collect 1 leaf, on the second day I collect 2 leaves, the third day I collect 3 leaves, and so on.
On my birthday, I will have collected 276 leaves altogether. Which day of the month is my birthday?
Bonus Question: if I keep collecting one more leaf each day, how many days would it take for me to collect 56,616 leaves?
Puzzle Copyright © Kevin Stone
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Hint
How many leaves will I have collected on day 5?
Answer
On the 23rd.
Reasoning
We could simply keep adding until we get the required number:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23
= 276 leaves.
But a more mathematical method might help to answer the Bonus Question, as this might take a while if we keep adding!
So, let's create a method by imagining that we are adding the numbers from 1 to 30.
1 + 2 + 3 + … + 28 + 29 + 30
If we now take the numbers in pairs, taking one from each end, we have:
(1 + 30) + (2 + 29) + (3 + 28) + … + (15 + 16)
Each pair adds to 31, and we have 15 pairs. So the total sum is 31 x 15 = 465.
The total sum from 1 to any number (N) can be found using this technique, and we will have:
Each pair adds to (1 + N), and there are N ÷ 2 pairs. So the total is:
(1 + N) x N
———
2
In this puzzle, we know that this equals 276.
So:
(1 + N) x N = 276
———
2
We can expand the brackets, and multiply both sides by 2, to give:
N + N2 = 552
Rearranging we get:
N2 + N − 552 = 0
And 552 = 2 x 2 x 2 x 3 x 23, so this can be factorised as:
(N + 24) x (N − 23) = 0
Because we need to find a positive number of days, the only possible answer is:
(N − 23) = 0
So N = 23 days.
Bonus Question
To answer the bonus question, we have:
(1 + N) x N = 56616
———
2
Rearranging we get:
N2 + N − 113232 = 0
And 113232 = 24 x 3 x 7 x 337, so this can be factorised as:
(N − 336) x (N + 337) = 0
Because we need to find a positive number of days, the only possible answer is:
(N − 336) = 0
So N = 336 days (I did say that I liked collecting leaves!).
Puzzle 200
Complete this grid of 4 four-letter words written horizontally.
Once complete, the first column of the grid forms a word, and the last column forms a word that is related to the first.
Note: this puzzle is not interactive, and the squares cannot be clicked.
Puzzle Copyright © Kevin Stone
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Hint
The first word starts with H.
Answer
Hard and soft.
| H |
A |
T |
S |
| A |
L |
S |
O |
| R |
O |
O |
F |
| D |
E |
B |
T |