Sunday Times Teaser 3334 – All Square

by BRG on 16 August 2026

by Victor Bryant

Published Sunday August 16 2026 (link)

The picture shows a grid of three rows and five columns of identical small squares: Within that grid one can also spot some larger squares, giving a grand total of 26 squares (namely fifteen 1×1 squares, eight 2×2 and three 3×3).

Now I have drawn a larger such grid (with fewer than 50 columns and even fewer rows).

Within this new grid there is a grand total of exactly 10,000 squares.

How many rows and how many columns does this new grid have?

From → Uncategorized

3 Comments Leave one →

BRG permalink


# number of squares in a rectangular M x N grid
# with M < N: M.(M + 1).(3.N - M + 1) / 6

for m in range(1, 50):
  d, r = divmod(60_000, m * (m + 1))
  if r == 0:
    n, r = divmod(m - 1 + d, 3)
    if r == 0 and n < 50:
      print(f"{m} rows by {n} columns")

# number of squares in a rectangular M x N grid

# with M < N: M.(M + 1).(3.N - M + 1) / 6

for m in range(1, 50):

d, r = divmod(60_000, m * (m + 1))

if r == 0:

n, r = divmod(m - 1 + d, 3)

if r == 0 and n < 50:

print(f"{m} rows by {n} columns")

Frits permalink


# after analysis: h * (h + 1) * (3 * w - h + 1) = 60000
# 60000 = 2^5 * 3 * 5^4
# (3 * w - h + 1) must be a multiple of 25 and 
# h or h + 1 must be a multiple of 25

for h in (24, 25):
  for k in range(3, 99):
    w, r = divmod(25 * k + h - 1, 3)
    if w > 49: break
    if r or h >= w: continue
    
    if h * (h + 1) * (3 * w - h + 1) == 60000:
      print(f"answer: {h} rows and {w} columns")

# after analysis: h * (h + 1) * (3 * w - h + 1) = 60000

# 60000 = 2^5 * 3 * 5^4

# (3 * w - h + 1) must be a multiple of 25 and

# h or h + 1 must be a multiple of 25

for h in (24, 25):

for k in range(3, 99):

w, r = divmod(25 * k + h - 1, 3)

if w > 49: break

if r or h >= w: continue

if h * (h + 1) * (3 * w - h + 1) == 60000:

print(f"answer: {h} rows and {w} columns")

Frits permalink

In one of my first attempts I used a recursive function (that can be cached) for the total.

Python

def f(w, h): if h == 1: return w else: return f(w, h - 1) + h * (2 * w - h + 1) // 2

1
2
3
4
5
6
7

def f(w, h):
if h == 1:
   return w
else:
   return f(w, h - 1) + h * (2 * w - h + 1) // 2

Reply

Sunday Times Teaser 3334 – All Square

by Victor Bryant

Published Sunday August 16 2026 (link)

Leave a Reply Cancel reply

Recent Posts

Recent Comments

Archives

Categories

Meta