Sunday Times Teaser 3208 – The Scores are All Square

by BRG on March 15, 2024

by Colin Vout

Published Friday March 15 2024 (link)

For Skaredahora’s quartet four players read the same musical score, but from different compass directions. There are symbols of three types, indicating different whole number beat durations, on a square 17×17 grid. Each player reads the beat position in their left-to-right direction, and pitch in their bottom-to-top.

Each player plays four notes; South reads a plus at (beat,pitch) position (3,12), a circle at (14,1), a cross at (16,3), and a plus at beat 9. For example, if a cross indicates three beats, South plays a note of pitch 3 at beat 16, which is still sounding at beats 17 and 18, while East plays a note of pitch 2 sounding at beats 3, 4 and 5.

No player sounds more than one note at the same time. All possible pitch differences between notes sounding simultaneously from different players occur, except zero and exactly one other value.

Which non-zero pitch difference never occurs? What pitch does South play at beat 9?

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BRG permalink


from itertools import combinations, product, zip_longest

# compose from four notes in <c4> and durations in <d3>
def compose(c4, d3):
  ds = dict(zip('+ox', d3))
  c = dict()
  for b, p, d in c4:
    s = {b + i: p for i in range(ds[d])}
    if s.keys() & c.keys():
      return
    c |= s
  return [0 if i not in c else c[i] for i in range(max(c) + 1)]
    
# accumulate non-zero pitch differences between the pitch sequences 
# <psq> for the four players when any of their notes overlap
def pitch_diff(psq):
  ds = set()
  # process the beats in the aligned pitch sequences ...
  for ps in zip_longest(*psq, fillvalue=0):
    # ... keeping pitch differences when any two player's notes overlap
    ds.update(abs(a - b) for a, b in combinations((p for p in ps if p), 2))
  # find non-zero pitch differences that don't occur
  r = set(range(1, 17)).difference(ds)  
  return r.pop() if len(r) == 1 and 0 not in ds else None 

# the unknoown value for the fourth note 
for p in range(1, 18):
  # the four different notes (beat, pitch, duration symbol) in S, W, N, E order
  bps = [[ ( 3, 12, '+'), (14,  1, 'o'), (16,  3, 'x'), (     9,      p, '+') ], 
         [ ( 6,  3, '+'), (17, 14, 'o'), (15, 16, 'x'), (18 - p,      9, '+') ], 
         [ (15,  6, '+'), ( 4, 17, 'o'), ( 2, 15, 'x'), (     9, 18 - p, '+') ], 
         [ (12, 15, '+'), ( 1,  4, 'o'), ( 3,  2, 'x'), (     p,      9, '+') ]]
  
  # find the maximum duration for each duration type
  mxd = {'+': 100, 'o': 100, 'x': 100}
  for p in bps:
    ds = sorted(p)
    for a, b in zip(ds, ds[1:]):
      mxd[a[2]] = min(mxd[a[2]], b[0] - a[0])
  if 0 in mxd.values():
    continue

  # consider possible durations for +, o and x
  for ds in product(*(range(1, mxd[s] + 1) for s in '+ox')):
    # the durations must be different
    if len(set(ds)) != 3:
      continue
    # store the four players pitch sequences
    psqs = [s for n4 in bps if (s := compose(n4, ds))]
    # look for sequences that have a single absent pitch
    # difference other than zero
    if (np := pitch_diff(psqs)):
      print(f"A pitch difference of {np} never occurs; "
            f"South's pitch at beat 9 is {psqs[0][9]}.")
      for n, s in zip('SWNE', psqs):
        print(f"{n}: {s}")

from itertools import combinations, product, zip_longest

# compose from four notes in <c4> and durations in <d3>

def compose(c4, d3):

ds = dict(zip('+ox', d3))

c = dict()

for b, p, d in c4:

s = {b + i: p for i in range(ds[d])}

if s.keys() & c.keys():

return

c |= s

return [0 if i not in c else c[i] for i in range(max(c) + 1)]

# accumulate non-zero pitch differences between the pitch sequences

# <psq> for the four players when any of their notes overlap

def pitch_diff(psq):

ds = set()

# process the beats in the aligned pitch sequences ...

for ps in zip_longest(*psq, fillvalue=0):

# ... keeping pitch differences when any two player's notes overlap

ds.update(abs(a - b) for a, b in combinations((p for p in ps if p), 2))

# find non-zero pitch differences that don't occur

r = set(range(1, 17)).difference(ds)

return r.pop() if len(r) == 1 and 0 not in ds else None

# the unknoown value for the fourth note

for p in range(1, 18):

# the four different notes (beat, pitch, duration symbol) in S, W, N, E order

bps = [[ ( 3, 12, '+'), (14, 1, 'o'), (16, 3, 'x'), ( 9, p, '+') ],

[ ( 6, 3, '+'), (17, 14, 'o'), (15, 16, 'x'), (18 - p, 9, '+') ],

[ (15, 6, '+'), ( 4, 17, 'o'), ( 2, 15, 'x'), ( 9, 18 - p, '+') ],

[ (12, 15, '+'), ( 1, 4, 'o'), ( 3, 2, 'x'), ( p, 9, '+') ]]

# find the maximum duration for each duration type

mxd = {'+': 100, 'o': 100, 'x': 100}

for p in bps:

ds = sorted(p)

for a, b in zip(ds, ds[1:]):

mxd[a[2]] = min(mxd[a[2]], b[0] - a[0])

if 0 in mxd.values():

continue

# consider possible durations for +, o and x

for ds in product(*(range(1, mxd[s] + 1) for s in '+ox')):

# the durations must be different

if len(set(ds)) != 3:

continue

# store the four players pitch sequences

psqs = [s for n4 in bps if (s := compose(n4, ds))]

# look for sequences that have a single absent pitch

# difference other than zero

if (np := pitch_diff(psqs)):

print(f"A pitch difference of {np} never occurs; "

f"South's pitch at beat 9 is {psqs[0][9]}.")

for n, s in zip('SWNE', psqs):

print(f"{n}: {s}")

Frits permalink

Following Jim’s set up.


from itertools import product, combinations

# return the player in the next anti-clockwise position
anti_clockwise = lambda seq: sorted([(b, 18 - a, s) for a, b, s in seq])

# return sequence of notes played by a specific player (zero if no note)
def notes_per_player(d_p, player):
  n = 17 + max(d_p.values()) - 1
  # extend a note
  d_ext = {b + j - 1: p for (b, p, s) in player for j in range(d_p[s])}
  return [0 if i not in d_ext else d_ext[i] for i in range(n)]

# return different pitch differences (zero not allowed)
gen_diffs = lambda bs: set(abs(p1 - p2) for i in range(len(bs[0]))
            for p1, p2 in combinations([b[i] for b in bs if b[i]], 2))

syms, s_players, n_notes = "ox+", "SENW", 4

# set up players without the unknown pitch coordinate
S = sorted([(3, 12, "+"), (14, 1, "o"), (16, 3, "x")])
E = anti_clockwise(S)
N = anti_clockwise(E)
W = anti_clockwise(N)

# add dummy plus entry at position 9
S = sorted(S + [(9, 0, "+")])
E = sorted(E + [(99, 9, "+")])
N = sorted(N + [(9, 0, "+")])
W = sorted(W + [(99, 9, "+")])

# determine maxima for symbols 'o', 'x' and '+'
d_mx = {sym: min([pl[i + 1][0] - b for pl in [S, E, N, W]
             for i, (b, _, s) in enumerate(pl)
             if sym == s and i < n_notes - 1]) for sym in syms}

# assume the unknown '+' at is at pitch <u> for South
for u in range(1, 18):
  # set up players
  S = [(3, 12, "+"), (9, u, "+"), (14, 1, "o"), (16, 3, "x")]
  E = anti_clockwise(S)
  N = anti_clockwise(E)
  W = anti_clockwise(N)
  players = [S, E, N, W]

  # choose combinations of durations for 'o', 'x', '+'
  for prod in product(*[range(1, d_mx[s] + 1) for s in syms]):
    # different whole number beat durations
    if len(set(prod)) != 3: continue

    d = dict(zip(syms, prod))
    # disallow simultaneous notes by the same player
    for pl in players:
      prev_ends = 0
      for b, p, s in pl:
        if b <= prev_ends: break
        prev_ends = b + d[s] - 1
      else:
        continue
      break
    else: # no break
      # determine notes played per beat
      notes = [beats_list for pl in players
               if (beats_list := notes_per_player(d, pl))]

      # calculate pitch differences
      diffs = gen_diffs(notes)
      if 0 in diffs: continue

      if len(diffs) == 15:
        print("answer: pitch difference that never occurs:",
              set(range(1, 17)).difference(diffs).pop())
        print("        South plays pitch", u, "at beat 9")

from itertools import product, combinations

# return the player in the next anti-clockwise position

anti_clockwise = lambda seq: sorted([(b, 18 - a, s) for a, b, s in seq])

# return sequence of notes played by a specific player (zero if no note)

def notes_per_player(d_p, player):

n = 17 + max(d_p.values()) - 1

# extend a note

d_ext = {b + j - 1: p for (b, p, s) in player for j in range(d_p[s])}

return [0 if i not in d_ext else d_ext[i] for i in range(n)]

# return different pitch differences (zero not allowed)

gen_diffs = lambda bs: set(abs(p1 - p2) for i in range(len(bs[0]))

for p1, p2 in combinations([b[i] for b in bs if b[i]], 2))

syms, s_players, n_notes = "ox+", "SENW", 4

# set up players without the unknown pitch coordinate

S = sorted([(3, 12, "+"), (14, 1, "o"), (16, 3, "x")])

E = anti_clockwise(S)

N = anti_clockwise(E)

W = anti_clockwise(N)

# add dummy plus entry at position 9

S = sorted(S + [(9, 0, "+")])

E = sorted(E + [(99, 9, "+")])

N = sorted(N + [(9, 0, "+")])

W = sorted(W + [(99, 9, "+")])

# determine maxima for symbols 'o', 'x' and '+'

d_mx = {sym: min([pl[i + 1][0] - b for pl in [S, E, N, W]

for i, (b, _, s) in enumerate(pl)

if sym == s and i < n_notes - 1]) for sym in syms}

# assume the unknown '+' at is at pitch <u> for South

for u in range(1, 18):

# set up players

S = [(3, 12, "+"), (9, u, "+"), (14, 1, "o"), (16, 3, "x")]

E = anti_clockwise(S)

N = anti_clockwise(E)

W = anti_clockwise(N)

players = [S, E, N, W]

# choose combinations of durations for 'o', 'x', '+'

for prod in product(*[range(1, d_mx[s] + 1) for s in syms]):

# different whole number beat durations

if len(set(prod)) != 3: continue

d = dict(zip(syms, prod))

# disallow simultaneous notes by the same player

for pl in players:

prev_ends = 0

for b, p, s in pl:

if b <= prev_ends: break

prev_ends = b + d[s] - 1

else:

continue

break

else: # no break

# determine notes played per beat

notes = [beats_list for pl in players

if (beats_list := notes_per_player(d, pl))]

# calculate pitch differences

diffs = gen_diffs(notes)

if 0 in diffs: continue

if len(diffs) == 15:

print("answer: pitch difference that never occurs:",

set(range(1, 17)).difference(diffs).pop())

print(" South plays pitch", u, "at beat 9")

Sunday Times Teaser 3208 – The Scores are All Square

by Colin Vout

Published Friday March 15 2024 (link)

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