Sample Task Kit for Home Lending

Answers: Secondary Kit

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At home with The Monkey
  • A Rectangle Of Squares

    1. The rectangle has a total area of 1,056 sq. cm.
    2. The rectangle is 33cm x 32cm.
    Extension
    We do not know of any other set of squares which create a rectangle. Nor can we prove that there is no other. You might be able to find another, or, you might be able to prove that there cannot be another set.

    Solution: A Rectangle of Squares

  • A Stacking Problem
    1. The puzzle takes 60 moves as follows:

    Action Moves Total So Far
    Blocks 1 & 2 to Position C. 3  
    Blocks 3 to Position B. 1  
    Blocks 1 & 2 back onto Block 3. 3 7 moves
    Block 4 to Position C. 1  
    Blocks 1 - 3 onto Block 4. 7 15 moves
    Block 5 to Position B. 1  
    Blocks 1 - 4 onto Block 5. 15 31 moves
    Block 6 to Position C. 1  
    Blocks 1 - 4 to Position A. 15 47 moves
    Block 5 onto Block 6. 1  
    Blocks 1- 3 to Position C. 7 55 moves
    Block 4 to Position B. 1  
    Blocks 1 & 2 to Position A. 3  
    Blocks 3 onto Block 4. 1 60 moves

    Extensions
    1. 5 moves
    2. 57 moves

  • Chess Queens
    There are several different solutions for each of the following situations.

    Seven Queens ... 7 queens
    Six Queens ... 6 queens
    Five Queens ... 5 queens

  • Dice Differences
    There is no specific answer to the game however, the theoretical frequency of the possible differences can guide placement strategy.

    Difference Escape rolls
    0 6 ways to escape
    1 10 ways to escape
    2 8 ways to escape
    3 6 ways to escape
    4 4 ways to escape
    5 2 ways to escape
      36 possible rolls

    Extension
    So far it is not certain which is the best placement strategy. Some possibilities are:

    1 2 2 1 0 0
    On average this takes about 13.7 rolls

    2 2 2 0 0 0
    On average this takes about 15.5 rolls

    1 3 2 0 0 0
    On average this takes about 14.4 rolls

  • Eric The Sheep
    If there are 50 in front of Eric to start with, 17 will be shorn before Eric gets to the front. Remember Eric's intention is to get to the front, so if there aren't two sheep to skip past when the shearer's back is turned, Eric will sneak past the one which is in the way.

    A table of results for different numbers of sheep shows an unusual pattern:

    Sheep in Front 1 2 3 4 5 6 7 8 9 ... 41 50 100 211
    Sheep shorn 1 1 1 2 2 2 3 3 3 ... 14 17 34 71

    • One rule is: Divide by 3 and round up if there is a remainder.
    • Another way to find the answer is expressed symbolically as: INT[(n - 1)/3] + 1
    • INT means to take the whole number (or integer) part of the an answer. So these symbols mean first subtract 1 from the number of sheep, then divide by 3, then use only the whole number part of the answer, and finally, add 1.
    • Another symbolic expression which gives the same result is: INT[(n + 2)/3]
    • Sneaking past three at a time becomes: INT[(n + 3)/4]
    • Sneaking past x at a time becomes: INT[(n + x)/(x + 1)]
    • With 2 shearers, and sneaking past 3 we get: 2 x INT[(n - 1)/(2 + 3)]

  • Eureka
    In total there are over 30 solutions. One is:
    • Peter: (5, 6, 9) = 20
    • Paul: (7, 8, 10, 15) = 40
    • Mary: (11, 14, 16, 17, 40) = 98
    but: How do we know when we have found them all?

    To find all the solutions one could systematically test all combinations for Peter and Paul from the lowest for Peter (5, 6, 7) = 18, to the highest for Paul (15, 16, 17, 40) = 88. Remember Paul's total must be twice that of Peter.

    A systematic list would look something like this:

    Peter Paul  
    (5, 6, 7) = 18 ( _, _, _, _ ) = 36 Lowest for Peter.
    (5, 6, 8) = 19 ( _, _, _, _ ) = 38  
    ... ...  
    ( _, _, _ ) = 44 (15, 16, 17, 40) = 88 Highest for Paul.

    Some results of this systematic search are:

    • Lowest:
      Peter (5, 6, 9) = 20, Paul (7, 8, 10, 15) = 40, Mary (11, 14, 16, 17, 40) = 98
    • Next Lowest:
      Peter (5, 6, 10) = 21, Paul (7, 8, 11, 16) = 42 or (7, 9, 11, 15) = 42
    • Highest:
      Peter (10, 14, 17) = 41, Paul (11, 15, 16, 40) = 82

  • First Down The Mountain
    The number of ways each total can be rolled, ie:

    Total 2 3 4 5 6 7 8 9 10 11 12
    No. of Ways 1 2 3 4 5 6 5 4 3 2 1

    seems to have been built into the symmetry of the board. We could reason that every climber has an equal chance of reaching the second last row, then, since Climber 7 has more chance of being rolled, it has more chance of reaching the winning line first.

    This intuitive reasoning seems reasonable, but it doesn't hold up when compared to the empirical data gained by playing the game many times. Climbers 2 and 12 win most often. How can this be explained?

    In fact, the explanation is not easy, but an indication of the reason comes by considering a race between the 2 and the 7, ignoring all other rolls, where the 2 has to go one step and the 7 has to go six steps.

    Race between 2 & 7
    From the table above, there are only 7 roll results which would be of interest. Six of these are wins for Climber 7 and only one is a win for Climber 2. So, Climber 7 would win 6/7 of the time. Climber 2 can win on any throw, but the only way Climber 7 can win is to roll 6 successive 7s before a single 2 is rolled. The theoretical chance of this is:
    6/7 x 6/7 x 6/7 x 6/7 x 6/7 x 6/7 = 39.6%.
    So in 1,000 trials we would expect Climber 7 to win 396 times which would mean Climber 2 would win 604 times.

  • Four In A Row (discontinued)
    To date, arrangements of counters have been found to give scores for all values up to 40 except 1, 2, 32, 35, 38 and 39. If you find arrangements for these points please let us know. Alternatively perhaps you could prove that one or more of these can't possibly be made.

    Examples
    This arrangement for one colour scores 28 points. 28 Points

    This arrangement for one colour scores 40 points. 40 Points

    This is one arrangement for getting zero points. Zero Points

  • Garden Beds
    1. 8       2. 10
    1. A table of results shows:
        Length of Bed (L) 1 2 3 4 5
        No. of Tiles (T) 8 10 12 14 16
    1. For any size bed the number of tiles is twice the length of the bed, plus another six. That is:
      T = 2 x L + 6
      It is twice because you need to surround both sides of the bed, then six extras allows for three at each end. If the bed was 100 units long, you would need 100 on one side, 100 on the other side, and 3 at each end. Therefore if N = 100, T = 206.
      There are other ways to 'see' the bed which will help you work out the number of tiles. Can you 'see' things a different way?
    Extensions
      Length of Bed (L) 23 38 100 247 1,396 40 140 982 5.5
      No. of Tiles (T) 52 82 226 500 2,798 86 286 1,970 17
  • Hearts & Loops
    The key to solving this puzzle is in seeing the hoop on the rod pass through the dip of the heart. It is something like a key in a lock image.

    Hearts & Loops Solution Place the central dip of the heart into this loop.

     

    Bring the large hoop(on the end of the rod) through the dip, from position A to position B.

     

    Lift the heart out of the loop to detach the pieces.

  • Highest Number 2
    • There are no specific answers to this task.
    • One analysis of a strategy, supposing the first two rolls were 5 and then 6, could be:
      5 is rolled first so place it in the centre position because it is in the middle of 1-9.

        5  

      The next roll is a 6. After that there are 7 numbers left for the third roll. If 6 is placed in the units, four of these (4, 3, 2, 1) would be a poor result, and only three (7, 8, 9) are good. But if the 6 is placed in the hundreds:

      6 5  

      The score will 650+ whatever the next roll.

      Possible scores the 1st way are:
      156, 256, 356, 456, 756, 856, 956 (Total = 3792)
      Possible scores the 2nd way are:
      651, 652, 653, 654, 657, 658, 659 (Total = 4584)
      Which decision is better in the long run?

  • Magic Cube
    The solution is shown here with the three vertical layers separated. The front one is on the left.
    6 is in the top left corner.
    14 is in the middle of the central layer.
    22 is on the bottom right corner.
    Together they make a line from front top left to rear bottom right which has a magic total of 42.

    Magic Cube Solution

    Calculating Magic Totals
    For a 4 x 4 x 4 cube made from 64 cubes:

    • Total from 1 to 64 = 65 x 32 = 2080
    • There are 16 distinct rows.
    • Magic Total = 2080 16 = 130
    For Size 5: Magic Total = 7875 25 = 315
    For Size 6: Magic Total = 23,436 36 = 651
    For Size N: Magic Total = [n/2 x (n + 1)] n = n/2 x (n + 1)

  • Mirror Patterns 2
    The results are:

    No. of Sides (N) 3 4 5 6 7 8 9 10
    Angle b/n Mirrors (A) 120 90 72 60 51.4 45 40 36

    The formula is N x A = 360.

    • If A = 100, then N = 3.6, ie: you can see 3.6 sides.
    • If A = 20, then N = 18, ie: you should be able to see 18 sides.

  • Protons & Anti-Protons
    • The answers to Question 1 are 2A, 1A and 3A
    • The finished chart is:

      + 3A 3P A P
      3P 0 6P 2P 4P
      2A 5A P 3A A
      P 2A 4P 0 2P
      A 4A 2P 2A 0

  • Red To Blue
    One approach, for even numbers of counters using the rule of turning all except 1, is:
    • Count consistently from the left (or right).
    • On move M, the Mth counter is unchanged and all others turn over.
    • Continue in order until all counters have been unchanged once and...

    Another approach is to draw a diagram like this:

    Red To Blue Tree Diagram
  • Snail Trail
    1. 7 hours
    2. In the 3rd hour (2 hours, 36 minutes)
    3. In the 6th hour (5 hours, 50 minutes, ie: it gets out more quickly.)
    4. (Up 3, Slip 2) or (Up 5, Slip 2) for example.

    Extensions

    1. 12 metres ... 5 hours
    2. 15 metres ... 6 hours (5 hours 30 minutes)
    3. 100 metres ... 49 hours (48 hours 24 minutes)
    4. In the 5th hour (4 hours 30 minutes)
    5. The important height (from which it can reach the top in the next hour) is 11 - x.
      It gains (x - y)/2 metres per hour.
      Hence it reaches 11 - x in (11 - x)/[(x - y)/2] hours, rounded up to the next hour.
      Then it reaches the top within the next hour, so the time is:
      (11 - x)/[(x - y)/2] - rounded up - plus 1.

  • Soma Cube 2
    There are no specific answers for this task.

  • Sphinx
    Size 2
    Sphinx Size 2
    requires 4 pieces.
    Size 3
    Sphinx Size 3
    requires 9 pieces.
    Size 4 is made using the Size 2 as a template and substituting a Size 2 for each Size 1 piece.
    Sphinx Size 4
    This requires 16 pieces.

    A Size 5 would require 25 pieces.
    Size 6 would need 36 pieces - use Size 3 as the unit and build a double Size 3.
    A Size N Sphinx would need NxN pieces. This shows the quadratic relationship between length and area.

  • Tower Of Hanoi
    • It takes 31 moves to relocate the tower made of 5 pieces.
    • For various numbers of pieces, the moves are:

      No. of Pieces 2 3 4 5 6 7 8 9 10 ... n
      No. of Moves 3 7 15 31 63 127 511 511 1023 ... 2n - 1

    • Notice that each answer is 'double the previous answer plus 1'.
      - First move the 'top' 3 pieces to another peg (7 moves).
      - Then move the base piece to the empty peg (1 move).
      - Then move the top 3 pieces back onto the base (7 moves).
    • Similarly, to move a tower of 10, first move the top 9 pieces (511) moves, then the base piece (1 move), then the top 9 back again (511 moves), for a total of 1023.
    • The formula m = 2n - 1 works for any number of discs.
      If n = 20, m = 1,048,575.
      Moving one disc per minute would take 104.02 weeks (or two years).
    • A tower of 100 pieces would take hundreds of billions of years.
      (2,400,000,000,000,000,000,000,000 years)

  • Truth Tiles 2

    • One solution is 3 + 7 - 4 = 6. Another is 6 + 3 - 5 = 4.
    • There are 12 solutions altogether.
      (Or 24 if you decide equations like 3 + 7 - 4 = 6 and 7 + 3 - 4 = 6 are different.)
    • Tiles 2, 3, 4, 5 and 6 give the same number of solutions.
    • One strategy is to recognise pair totals such as 9 = (3 + 6) or (4 + 5).
      Arrangements of these pairs gives 8 solutions, ie: 3 + 6 - 4 = 5
      6 + 3 - 4 = 5
      3 + 6 - 5 = 4
      6 + 3 - 5 = 4
      4 + 5 - 3 = 6
      5 + 4 - 3 = 6
      4 + 5 - 6 = 3
      5 + 4 - 6 = 3
    • Similarly, the pair totals of 10 = (3 + 7) or (4 + 6) and 11 = (4 + 7) or (5 + 6) also give 8 solutions each.
    • Using 6 numbers gives a total of 28 combinations.
    • Using 7 numbers gives 52 combinations.
    • Using 8 numbers gives 88 combinations.

  • Win At The Fair
    1. The most frequent result is 50. If you reached the position immediately below the 20 prize, the chance of going left or right is 20/36, compared to going ahead (16/36).
    2. The largest prize ($5) is placed where the same rolls are needed one after the other, ie: LLL or RRR. Theoretically, the chance of each of these is:
      10/36 x 10/36 x 10/36 = 2.14%
      Hence 4.28% of players can expect to win $5.
    3. The operator in this setup would LOSE money.
      For 1000 players, the money in is $1000.
      The payout is $1,263.40 for a LOSS of $263.40 or 26.3%.
      If, for example, the $1 prize was changed to NIL, then the operator would just break even.

    Extension
    There are six pathways to win $4 on either side, hence 12 altogether, eg: RLLL or LULLL.