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Posers and Puzzles

Posers and Puzzles

  1. 02 Nov '11 09:24 / 1 edit
    yes the shame of it, spent ages last night trying to solve this and admitted defeat, i
    would be very pleased if anyone could help me. The question is as follows.

    join the numbers horizontally or vertically so that you make a continuous path
    connecting twelve numbers that add up to 174


    16 12 17 11 15
    13 18 10 14 12
    17 11 18 11 16
    10 17 12 16 10
    16 12 15 13 17


    kind regards Robbie.
  2. Standard member talzamir
    Art, not a Toil
    02 Nov '11 10:11 / 2 edits
    I imagine this is a real nightmare to do with pen and paper only, but very easy with a computer. I used Excel, copied the numbers there in cells a1..e5, and made the equation

    =sum(a1:e5)-174

    nearby. Originally that gives 175 as the sum of the 25 numbers is 349 and 349-174 = 175. Next I started deleting numbers from the box until I ended up with zero, avoiding untraceable paths such as isolated numbers or three or more with only one neighbor. Once the equation gives a zero the sum of what is left is 174. If it is positive, delete some more; if negative, replace something and adjust. ctrl-z helps nicely. Here are a few of the prettier solutions I came up with.


    12-10-17-13-16-12-17-11-11-15-12-16-10-13
    A symmetrical horse shoe that starts and ends on the bottom row. Goes along the entire edge of the box, except for the bottom corners of the middle of the bottom row.

    17-12-14-10-18-18-17-12-12-16-13-15
    This one starts and ends at the top and bottom of the middle column. It goes through everything, except the leftmost and the rightmost column, and avoids all 11's.

    10-17-11-15-12-16-10-17-13-15-12-16-10
    My favorite. There are three 10's in the box and in this solution they are at the start, middle, and end of the line. The pattern doesn't use diagonal moves at all - it's a spiral that starts at the center and goes clockwise along the edge.

    10-12-12-13-10-16-11-14-12-11-10-12-13-18
    There are no solutions with diagonal moves only, but this is pretty close as b3 is the only"white square" missing. Start near the lower left corner, zigzag right, up along the d-column, zigzag left, and end up in b4.

    There are heaps of other solutions if you wish to find your own. I hope this helps.
  3. Standard member talzamir
    Art, not a Toil
    02 Nov '11 12:06 / 3 edits
    Sorry about that. I should read more carefully. Exactly 12 numbers. And no diagonals. All but one of mine have diagonals, and the spiral has 13 numbers, so back to the drawing board it is.

    If 12 numbers form a sum of of 174, then the average of the 12 numbers is 14.5. To avoid fractions, I doubled all the numbers, so the average is 29. I then subtracted 29 from each, and sought a path of 12 numbers with a sum of zero. Excel can paint all negatives red and positive black with a single click which also helped.


    +3 -5 +5 -7 +1
    -3 +7 -9 -1 -5
    +5 -7 +7 -7 +3
    -9 +5 -5 +3 -9
    +3 -5 +1 -3 +5

    And that is fairly easy to see as there are lots of pairs of adjacent numbers with the same absolute value but a different sign. Pick both for a sum of zero. In other words, I sought for six pairs of numbers where the sum of the two numbers is 29 as 6 x 29 = 174, and found one.

    In chess terms, the path I found is on an A1..E5 board,

    D1-D2-D2-C3-C2-B2-B3-B4-A4-A5-B5-C5.

    (13+16)+(11+18)+(12+17)+(11+18)+(13+16)+(12+17)
    = 29+29+29+29+29+29
    = 6x29
    = 174.


    That's probably the one that the teacher who made the puzzle had in mind. There are of course others, such as

    E1-D1-C1-B1-B2-C2-B3-B4-B5-A5-A4.
    17+13+15+12+17+12+18+11+18+12+16+13 = 174.
  4. 02 Nov '11 16:16
    Originally posted by talzamir
    Sorry about that. I should read more carefully. Exactly 12 numbers. And no diagonals. All but one of mine have diagonals, and the spiral has 13 numbers, so back to the drawing board it is.

    If 12 numbers form a sum of of 174, then the average of the 12 numbers is 14.5. To avoid fractions, I doubled all the numbers, so the average is 29. I then subtracted ...[text shortened]... hers, such as

    E1-D1-C1-B1-B2-C2-B3-B4-B5-A5-A4.
    17+13+15+12+17+12+18+11+18+12+16+13 = 174.
    thankyou so much.
  5. Standard member Phlabibit
    Mystic Meg
    02 Nov '11 20:20
    Originally posted by robbie carrobie
    yes the shame of it, spent ages last night trying to solve this and admitted defeat, i
    would be very pleased if anyone could help me. The question is as follows.

    join the numbers horizontally or vertically so that you make a continuous path
    connecting twelve numbers that add up to 174


    16 12 17 11 15
    13 18 10 14 12
    17 11 18 11 16
    10 17 12 16 10
    16 12 15 13 17


    kind regards Robbie.
    Congratulations on your kid getting into Harvard.

    P-
  6. 02 Nov '11 23:12
    Originally posted by Phlabibit
    Congratulations on your kid getting into Harvard.

    P-
    LOL, this is primary six homework, age ten! The shame of not being able to solve it.
    My kid will be doing well to write his name when he leaves school, hes a brilliant kid,
    just not very brainy!
  7. 04 Nov '11 00:01 / 1 edit
    There are 27 ways of traversing the grid to get 174, counting only those that go through unique combinations of cells:
    First row is indices, next row is values added up.

    Indices are row_col, top down, left to right, so 0_0 is top left.

    0_0 0_1 1_1 1_0 2_0 2_1 3_1 3_2 2_2 2_3 3_3 4_3
    16 +12 +18 +13 +17 +11 +17 +12 +18 +11 +16 +13 = 174

    0_0 0_1 1_1 1_0 2_0 2_1 2_2 3_2 3_1 4_1 4_2 4_3
    16 +12 +18 +13 +17 +11 +18 +12 +17 +12 +15 +13 = 174

    0_1 1_1 1_0 2_0 3_0 3_1 2_1 2_2 3_2 3_3 4_3 4_4
    12 +18 +13 +17 +10 +17 +11 +18 +12 +16 +13 +17 = 174

    0_1 1_1 1_0 2_0 3_0 3_1 3_2 2_2 2_3 3_3 4_3 4_4
    12 +18 +13 +17 +10 +17 +12 +18 +11 +16 +13 +17 = 174

    0_1 1_1 1_0 2_0 2_1 3_1 4_1 4_2 4_3 3_3 3_2 2_2
    12 +18 +13 +17 +11 +17 +12 +15 +13 +16 +12 +18 = 174

    0_1 0_0 1_0 1_1 2_1 3_1 3_2 2_2 2_3 3_3 4_3 4_4
    12 +16 +13 +18 +11 +17 +12 +18 +11 +16 +13 +17 = 174

    0_1 0_0 1_0 1_1 2_1 2_2 3_2 3_1 4_1 4_2 4_3 4_4
    12 +16 +13 +18 +11 +18 +12 +17 +12 +15 +13 +17 = 174

    0_1 0_2 1_2 1_1 1_0 2_0 2_1 2_2 3_2 3_3 4_3 4_4
    12 +17 +10 +18 +13 +17 +11 +18 +12 +16 +13 +17 = 174

    0_2 0_1 1_1 1_0 2_0 3_0 3_1 2_1 2_2 3_2 3_3 4_3
    17 +12 +18 +13 +17 +10 +17 +11 +18 +12 +16 +13 = 174

    0_2 0_1 1_1 1_0 2_0 3_0 3_1 3_2 2_2 2_3 3_3 4_3
    17 +12 +18 +13 +17 +10 +17 +12 +18 +11 +16 +13 = 174

    0_2 0_1 1_1 1_0 2_0 2_1 3_1 3_2 2_2 2_3 2_4 1_4
    17 +12 +18 +13 +17 +11 +17 +12 +18 +11 +16 +12 = 174

    0_2 0_1 1_1 1_0 2_0 2_1 2_2 3_2 3_3 4_3 4_2 4_1
    17 +12 +18 +13 +17 +11 +18 +12 +16 +13 +15 +12 = 174

    0_2 0_1 1_1 1_0 2_0 2_1 2_2 3_2 3_3 4_3 4_4 3_4
    17 +12 +18 +13 +17 +11 +18 +12 +16 +13 +17 +10 = 174

    0_2 0_1 1_1 1_0 2_0 2_1 2_2 2_3 3_3 3_2 3_1 4_1
    17 +12 +18 +13 +17 +11 +18 +11 +16 +12 +17 +12 = 174

    0_2 0_1 0_0 1_0 1_1 2_1 3_1 3_2 2_2 2_3 3_3 4_3
    17 +12 +16 +13 +18 +11 +17 +12 +18 +11 +16 +13 = 174

    0_2 0_1 0_0 1_0 1_1 2_1 2_2 3_2 3_1 4_1 4_2 4_3
    17 +12 +16 +13 +18 +11 +18 +12 +17 +12 +15 +13 = 174

    0_3 0_2 0_1 1_1 1_0 2_0 2_1 3_1 3_2 3_3 4_3 4_4
    11 +17 +12 +18 +13 +17 +11 +17 +12 +16 +13 +17 = 174

    0_3 0_2 0_1 1_1 1_0 2_0 2_1 2_2 3_2 4_2 4_3 4_4
    11 +17 +12 +18 +13 +17 +11 +18 +12 +15 +13 +17 = 174

    0_3 0_2 0_1 1_1 1_0 2_0 2_1 2_2 3_2 3_1 4_1 4_0
    11 +17 +12 +18 +13 +17 +11 +18 +12 +17 +12 +16 = 174

    0_3 0_2 0_1 1_1 1_0 2_0 2_1 2_2 2_3 3_3 4_3 4_4
    11 +17 +12 +18 +13 +17 +11 +18 +11 +16 +13 +17 = 174

    0_3 0_2 0_1 0_0 1_0 1_1 2_1 2_2 3_2 3_3 4_3 4_4
    11 +17 +12 +16 +13 +18 +11 +18 +12 +16 +13 +17 = 174

    1_0 1_1 0_1 0_2 1_2 2_2 3_2 3_1 4_1 4_2 4_3 4_4
    13 +18 +12 +17 +10 +18 +12 +17 +12 +15 +13 +17 = 174

    1_0 1_1 0_1 0_2 1_2 2_2 2_1 3_1 3_2 3_3 4_3 4_4
    13 +18 +12 +17 +10 +18 +11 +17 +12 +16 +13 +17 = 174

    2_0 1_0 1_1 0_1 0_2 1_2 2_2 3_2 3_1 4_1 4_2 4_3
    17 +13 +18 +12 +17 +10 +18 +12 +17 +12 +15 +13 = 174

    2_0 1_0 1_1 0_1 0_2 1_2 2_2 2_1 3_1 3_2 3_3 4_3
    17 +13 +18 +12 +17 +10 +18 +11 +17 +12 +16 +13 = 174

    4_1 3_1 2_1 2_0 1_0 1_1 1_2 2_2 3_2 3_3 4_3 4_4
    12 +17 +11 +17 +13 +18 +10 +18 +12 +16 +13 +17 = 174

    4_1 3_1 3_0 2_0 1_0 1_1 2_1 2_2 3_2 3_3 4_3 4_4
    12 +17 +10 +17 +13 +18 +11 +18 +12 +16 +13 +17 = 174
  8. Standard member talzamir
    Art, not a Toil
    04 Nov '11 07:21
    wow...!