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# Clock puzzles

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### Clock puzzles

1. 1. CLOCK PUZZLES
2. 2. CLOCK PUZZLES Answer 5:19 Explanation Starting with the watch on the left, add 42 minutes to the time shown to give the time on the next watch to the right.
3. 3. CLOCK PUZZLES Hand pointing to 5 Explanation Starting with the top clock and moving anti- clockwise around the others, the hour hand moves back 1 hour, then 2, then 3 etc, while the minute hand moves forward 10 minutes each time.
4. 4. CLOCK PUZZLES To the 4 Explanation Start with the top left clockface, and move around the others in a clockwise direction. The value pointed to by the hour hand equals the value pointed to by the minute hand, subtracting 4 for the first clockface, 5 for the next, then 6 etc.
5. 5. CLOCK PUZZLES Answer Hour hand is pointing to 2. Explanation :Multiply the hour hand value by 2 and add the minute hand value. This total is always 15.
6. 6. CLOCK PUZZLES Answer: To the 5 Explanation Start with the top left clockface, and move around the others in a clockwise direction. The sum of the numbers pointed to by the hour and minute hand follows the sequence 14, 15, 16, 17 and 18.
7. 7. CLOCK PUZZLES Answer : 11:01 Explanation : Taking the hour and minute values on each watch separately, as you move to the right the hour value increases by 1, 2, 3 and 4, and the minute value decreases by 11, 22, 33 and 44.
8. 8. PLAYING CARDS
9. 9. Answer: 9 of Clubs Explanation: Taking red cards as positive values and black cards as negative values, in each column of the diagram, the lower card value equals the sum of the two upper card values. The suits are used alternately in each column.
10. 10. Answer: King of Clubs Explanation Start at the top left of the diagram and move to the right, then down one row and to the left etc. in a snakes and ladders pattern. The value of each card increases by 5 each time, with their suit following the sequence of hearts, clubs, diamonds and spades.
11. 11. Answer 7 (any suit) Explanation : Taking the value of Aces as 1 and all court cards as 10, In each column of the diagram, the value of the sum of the 3 cards is always 21.
12. 12. Answer : Jack of SpadeS Explanation : There are 2 sequences in the grid - one determining the value of the card, and one determining the suit of the card. Starting on the top left and moving right, then down one row and to the left, then down the final row and to the right, cards are arranged in order, with their value increasing by 4 each time. To calculate the suit of each card, start on the top left and move down, then right one row and move up etc. cards are arranged in the order Hearts, Clubs, Diamonds, Spades.
13. 13. Answer : 3 of SpadeS Explanation : Divide the diagram in half, vertically. In each half, start at the top left card and move to the right, then down one row and to the left, and finally to the right, in a snakes and ladders pattern. The value of the cards in the left hand half increase alternately by 3 and 4, and the value of the cards in the right hand half increase alternately by 4 and 5. To calculate the suit of each card, start at the top left of the whole diagram and move down, then to the right one space and upwards etc. in a snakes and ladders pattern. Suits are written in order, following this path, starting with Hearts, then Clubs, Diamonds and Spades.
14. 14. LOGICAL PUZZLES Answer : D Explanation The number of black dots in each grid increases by 1 each time, starting with the top left grid and working to the right, top row then bottom row. Answer: Q Explanation: Adding the three numbers in each square together gives the numerical value of the letter at the centre of each square.
15. 15. Answer : G Explanation Convert each letter to its numerical value, and read each pair of values as 2 digit numbers. In each row, the number in the centre equals the difference between the 2 digit values on the left and right. Answer : E Explanation In each row, the left hand grid is symmetrical around a vertical axis, the central grid is symmetrical about a horizontal axis, and the right hand grid is symmetrical about a diagonal axis, running bottom left to top right.
16. 16. Answer :Hammer Explanation :The Hammer = 1, the File = 3 and the Axe = 5 Answer:U Explanation: Multiply the numerical values of the letters in each pair to give the 3 digit result in the spaces above.
17. 17. Answer:12 Explanation :The value at each corner of the diagram equals the difference between the sums of the numerical values of the letters in the boxes adjacent to the corner. Answer: C
18. 18. Answer P Explanation:Working in rows, add the left and right hand numbers together, and put the letter with the reverse alphabetical value of this sum in the central box. Answer:4 Explanation : Split the left and right hand circles in half vertically. The letter with the numerical value of the sum of the digits in the left half of the left hand circle is placed in the top left segment of the central circle, and the letter with the numerical value of the sum of the digits in the right half of the left hand circle is placed in the top right segment of the central circle. Repeat this formula for the 2 halves of the right hand circle, putting the resulting letters in the lower segments of the central circle.
19. 19. MISSING LETTER PUZZLES Answer M Explanation Working in rows, add together the numerical values of the left and right hand letters to give the numerical value of the central letter. Answer O Explanation In each segment of the diagram are a pair of letters, one of which is the same distance from the start of the alphabet as the other is from the end.
20. 20. MISSING LETTER PUZZLES Answer : K Explanation : The numerical values of the letters in opposite segments of the circle always add up to 17. Answer:K Explanation: As you move down, the numerical value of the letters follows the sequence of Prime Numbers. Answer : 1 Explanation :Working from left to right, letters in corresponding segments of the circles move through the alphabet in steps of 2, 3 and 4, with their relative positions moving one place clockwise at each step.
21. 21. MISSING LETTER PUZZLES Answer :P Explanation : Starting at the top left circle, and moving right, then down one row and moving left, in a snakes and ladders pattern, letters move through the alphabet in steps of 2, 3 and 4, repeating this pattern all the way Answer:2: Splitting the large square into quarters, each quarter features the same pattern of letters. Working from left to right, letters move forwards 7 places, then back 2. Repeat this sequence until the end. ANS : W
22. 22. NUMBER PUZZLES Taking the top row of circles, numbers in the central circle equal the sum of the numbers in corresponding segments of the left and right hand circles. In the bottom row, numbers in the central circle equal the difference between numbers in corresponding segments of the left and right hand circles. ANS: 7 Starting with the 10 at the top, one set of numbers increases by 3 each time, written in alternate boxes as you move down the diagram, and the other set of numbers decreases by 2, written in the boxes remaining.ans 2
23. 23. NUMBER PUZZLES Starting with the top 3 digit number, the first digit increases by 2 as you descend, from 1 to 11. The middle digit decreases by 1 each time, and the right hand digit alternates between 5 and 3. Reading each row of the diagram as a series of 3 digit numbers, the centre 3 digit number equals the sum of the top 2 numbers, and the sum of the lower 2 numbers. Add together values in corresponding positions of the top two crosses, and put the results in the lower left cross. Calculate the difference between values in corresponding positions of the top two crosses, and put the results in the lower right cross. Finally, add together the values in corresponding positions of the lower two crosses to give the values in the central cross.
24. 24. NUMBER PUZZLES In each row of the diagram, the central value equals the sum of the differences between the left hand pair of numbers and the right hand pair of numbers. Ans 7 The inner digit in each segment equals the sum of the two numbers in the outer part of the opposite segment. Ans 7
25. 25. SUDOKU Crosshatching - finding squares for numbers. The obvious way to solve a sudoku puzzle is to find the right numbers to go in the squares. However the best way to start is the other way round - finding the right squares to hold the numbers. This uses a technique called 'crosshatching', which only takes a couple of minutes to learn. It can solve many 'easy' rated puzzles on its own.
26. 26. SUDOKU Crosshatching works in boxes (the 3 X 3 square subdivisions of the grid). Look at the top- left box of our sample puzzle (outlined in blue). It has five empty squares. All the numbers from 1 to 9 must appear in the box, so the missing numbers are 1,2,3,5 and 9. We'll ignore 1 for a moment (because it doesn't provided a good example!), and see if we can work out which square the missing 2 will go into.
27. 27. SUDOKU To do this, we'll use the fact that a number can only appear once in any row or column. We start by looking across the rows that run through this box, to see if any of them already contain a 2. Here's the result: The first two rows already contain 2s, which means that squares in those rows can't possibly contain the 2 for this box. That's all we need to know, because the third row only has one empty square, so that must be the home for the 2. Now let's see if we can place the 3 for this box. This time we end up checking the columns that run down through the box, as well as the rows that run across it:
28. 28. SUDOKU Again, we get a result first time – there's only one empty square that the 3 can possibly go into. You can see from this example why it's called 'crosshatching' - the lines from rows and columns outside the square criss-cross each other. Of course you don't always get a result first time. Here's what happens when we try to place the 5:
29. 29. SUDOKU There's only one 5 already in the rows and columns that run through this box. That leaves three empty squares as possible homes for the 5. For the time being, this box's 5 (and its 1 and 9) have to remain unsolved. Now we move on to the next box: Here we're crosshatching for 3, the first missing number in this box. Note how we treat the 3 we placed in the first box as if it had been pre-printed on the puzzle. We still can't place this box's 3 though, so we'll move on to the next missing number (5), and so on. In sudoku, accuracy is essential. If the 3 in the first box is wrong, we'll be starting a chain of errors that may prove impossible to unravel. Only place a number when you can prove, logically, that it belongs there. Never guess, and never follow hunches!
30. 30. SUDOKU In sudoku it pays to look at the same thing in different ways. By using crosshatching slightly differently, you can often get quicker results. Instead of looking at a single box and its missing numbers, you can look at a group of three boxes running across or down the puzzle, trying to place each number from 1 to 9 in as many of the boxes as you can. In this example we're trying to place 7s in the three right-hand boxes: The stack of boxes starts out with just one 7 in place (bottom box). This solves the middle box's 7 (entered in blue), and entering that immediately solves the top box's 7 as well. This 'chain reaction' of solving wouldn't occur in single-box crosshatching. It happens here because we're focussing on a single number across multiple boxes - looking at things differently. Crosshatching and slicing/dicing are basically the same thing, but slicing/dicing can be more efficient, and often feels less laborious than doggedly working through the empty squares in a single box (although that's what you will have to do in order to solve tough puzzles, so be prepared!).
31. 31. SUDOKU Looking at the top-left box of our original puzzle, crosshatching produced three possible squares where the missing 5 could go. Here's the box, with 5 'pencilled-in' to the corners of its three possible squares: 1 and 9 were also unsolved for this box. Here's the box with all its missing numbers pencilled into their possible squares: On the left is is the top-left box, and the one below it. We've just entered a 5 in the bottom- right square of the lower box. Now we remove that square's candidate list. We also remove 5 from the candidate list at the top of the same column, and the left of the same row. Here's how the boxes look afterwards:
32. 32. SUDOKU Whenever you fill in a square, remove the number you've used from all candidate lists in the same row, column and box. Here are the areas we needed to check for candidate 5s after filling in this square:
33. 33. SUDOKU Rule 1 - Single-candidate squares. When a square has just one candidate, that number goes into the square. Here's the mid-left box again, as it was before we entered the 5: The mid-right and bottom-right squares each have only one candidate, so we can put those numbers into the squares. Some squares will be single-candidate from the start of the puzzle. Most, however, will start with multiple candidates and gradually reduce down to single-candidate status. This will happen as you remove numbers that you've placed in other squares in the same row, column and box, and as you apply the last three rules described below. Rule 2 - single-square candidates. When a candidate number appears just once in an area (row, column or box), that number goes into the square. Look at the mid-left box again: The number 6 only appears in one square's candidate list within this box (top- middle). This must, therefore, be the right place for the 6. The remaining three rules let you remove numbers from candidate lists, reducing them down towards meeting one of the first two rules.
34. 34. SUDOKU Rule 3 - number claiming. When a candidate number only appears in one row or column of a box, the box 'claims' that number within the entire row or column. Here's the top-left box again: The number 1 only appears as a candidate in the top row of the box. This means there will have to be a 1 somewhere in the first three squares of the puzzle's first row (i.e. the ones that overlap with the box). That in turn means that 1 can't go anywhere else in that row, outside of this box. You can read across the row, and remove 1 from any candidate lists outside of this box, even though you haven't actually placed 1 yet. In this example, we can remove the 1 from the right-hand square's candidate list. This makes the square single-candidate (7) - square solved!
35. 35. SUDOKU Claims also work during crosshatching. Here we're crosshatching the top-right box for 1: We can rule out the top row, because the top- left box has already claimed that row's 1. This lets us place the 1 in the bottom-right square of the box.
36. 36. SUDOKU Rule 4 - pairs. When two squares in the same area (row, column or box) have identical two- number candidate lists, you can remove both numbers from other candidate lists in that area. Here's the second row of the puzzle: Two of the squares have the same candidate list - 67. This means that between them, they will use up the 6 and 7 for this row. That means that the other square can't possibly contain a 6. We can remove the 6 from its candidate list, leaving just 9 - square solved! The squares in a pair must have exactly two candidates. If one of the above squares had been 679, it couldn't have been part of a pair.
37. 37. SUDOKU Rule 5 - triples. Three squares in an area (row, column or box) form a triple when: None of them has more than three candidates. Their candidate lists are all full or sub sets of the same three-candidate list (explained below!). You can remove numbers that appear in the triple from other candidate lists in the same area. Here's the fourth row of the puzzle: Note the three squares in the middle, with candidates of 23, 23 and 234. These form a triple. 234 is the full, three-candidate list, and 23 is a subset of it (i.e. all its numbers appear in the full list). Because there are three squares, and none of them have any candidate numbers outside of those in the three-candidate list, they must use up the three candidate numbers (2, 3 and 4) between them. This lets us remove the 4 from the other two candidate lists in this row, solving their squares.
38. 38. SUDOKU It's worth looking hard for subset triples. In this example, the 23 lists make an obvious pair (see above), but it's the triple that instantly solves the two outside squares (once you've dispensed with them, you can treat the 23s as a pair again, and use them to solve the 234!). A subset (or 'hidden') triple is often the pattern that will unlock a seemingly impossible puzzle. Note - the squares in a pair or triple don't have to appear next to each other, or in any particular order. In the example above, the triple could have occurred in, say, the first, third and fifth empty squares of the row, with the 234 in the middle.
39. 39. SUDOKU • the triple rule can be true even if none of the squares have three candidates. Take these three candidate lists: • 13 16 36 • All three lists are subsets of the list 136. Between them, these three squares will use up the 1, 3 and 6 for the area they're in. These triples can be hard to spot though, so it's probably best to start by looking out for three-candidate squares. • (Special thanks to Edward for pointing out that the members of a triple can all be subsets of the full list!) • ◊ If all-subset triples still don't seem right, think of it this way: The crucial thing is that the number of squares equals the number of candidates in the full list (so three squares all with subsets of '136' (a three-candidate list) makes a triple). It doesn't matter if some (or all) of the squares don't have the full candidate list. What matters is that between them they cover the list, the whole list and nothing but the list. That means they must use up all three of the list's numbers between them.
40. 40. SUDOKU In case all that's put you off, here's an example of a more obvious triple - they do exist!: A set of N squares in an area forms a group when: None of the squares has more than N candidates Their candidate lists are all full or sub sets of the same N-candidate list. Pairs (N=2) with subset members tend not to survive long, because a subset of a two-number list is a single candidate and thus solvable. However a single-candidate square can function perfectly well as a member of a pair, or even a triple. This often has the same effect as solving the single-candidate square then updating its surrounding candidate lists, but can be quicker. On the left is an example (and from an 'easy' rated puzzle, too!) The top three unsolved squares, with candidates of 26, 23 and 2, form a triple (N=3) with a full list of 236. That lets us remove the 3 and 6 from the bottom square's candidate list, reducing it to just 8 - square solved! You also sometimes see 'quadruplets' (N=4) - four squares, none with more than four candidates, and all full or subsets of a four-candidate list.