Sunday Times Teaser 2924 – Snaking Up

by Victor Bryant

Published October 7 2018 (link)

A “Snakes and Ladders” board consists of a 10-by-10 grid of squares. In the first row (at the bottom) the squares are numbered from 1 to 10 from left to right, in the second row the squares are numbered 11 to 20 from right to left, in the third they are 21 to 30 from left to right again, and so on.

An ant started on square 1, moved to square 2 and then, always moving to an adjacent square to the right or up, it finished in the top-right corner of the board. I have added up the total of the numbers on the squares it used and, appropriately, the total is a perfect square. In fact it is the square of one of the numbers the ant visited — the ant passed straight through it without turning.

What was that total of the numbers used?

4 Comments Leave one →

Brian Gladman permalink


# find a path through a 10 by 10 grid, starting at (0, 0) and
# ending at (9, 9) with moves that increment either x  or y but
# not both; return the 'Snakes and Ladders' numbers on the path
def solve(x, y, seq):
  # compute the 'Snakes and Ladders' number from (x, y)
  n = 10 * y + (10 - x if y % 2 else x + 1)
  s = seq + (n,)
  if x == 9 and y == 9:
    yield s
  else:
    if x < 9:
      yield from solve(x + 1, y, s)
    if y < 9:
      yield from solve(x, y + 1, s)

# check all valid paths across the board (the first
# move is specified)
for seq in solve(1, 0, (1,)):
  # check that the sum of the numbers on the path
  # is the perfect square of a number it contains
  sm = sum(seq)
  sr = int(sm ** 0.5)
  if sr ** 2 == sm and sr in seq:
    # check that the path through this number
    # doesn't change direction
    ix = seq.index(sr)
    if seq[ix + 1] - seq[ix - 1] in {2, 20}:
      print(f'sum = {sm} ({sr}^2) {seq}')

# find a path through a 10 by 10 grid, starting at (0, 0) and

# ending at (9, 9) with moves that increment either x or y but

# not both; return the 'Snakes and Ladders' numbers on the path

def solve(x, y, seq):

# compute the 'Snakes and Ladders' number from (x, y)

n = 10 * y + (10 - x if y % 2 else x + 1)

s = seq + (n,)

if x == 9 and y == 9:

yield s

else:

if x < 9:

yield from solve(x + 1, y, s)

if y < 9:

yield from solve(x, y + 1, s)

# check all valid paths across the board (the first

# move is specified)

for seq in solve(1, 0, (1,)):

# check that the sum of the numbers on the path

# is the perfect square of a number it contains

sm = sum(seq)

sr = int(sm ** 0.5)

if sr ** 2 == sm and sr in seq:

# check that the path through this number

# doesn't change direction

ix = seq.index(sr)

if seq[ix + 1] - seq[ix - 1] in {2, 20}:

print(f'sum = {sm} ({sr}^2) {seq}')

Benjamin Cooper permalink

Out of 3 potential candidates (676, 784 and 1225),

I find only 784 (28^2) meets solution requirements, passing through 1,2,12,22,23,24,25,26,27,28,29,39,49,59,69,79,80,90,100

Lowest scoring route is anticlockwise around perimeter. Each “deviation” exposing a square, worth 9 added each. The white paper squares are not all adjacent..

Reply
- Brian Gladman permalink
  
  Hi Ben, Your path doesn’t work since the number above 2 is 19, not 12. There are a number of paths with a sum of 784 (28^2) but all of them involve a change of direction when passing through 28. It looks as if you are assuming that all rows are numbered left to right whereas the order alternates between left to right and right to left.
  
  Reply
Benjamin Cooper permalink

Thanks Brian – too much haste!

Reply

Sunday Times Teaser 2924 – Snaking Up

by Victor Bryant

Published October 7 2018 (link)

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