Leilani placed a digit in each of these 16 squares:
+------+------+------+------+
| | | | |
+---- -+------+------+------+
| | | | |
+------+------+------+------+
| | | | |
+------+------+------+------+
| | | | |
+------+------+------+------+
When she had finished, the grid had the following properties:
• No digit occured more than once in any row.
• The sum of the four digits in each row was the same.
• The sum of the four digits in each column was the same.
• Each row formed a different four-digit perfect square.
Complete the 4x4 square grid given above.
Note: Adapted from Enigma #1553 which appeared in 'New Scientist' in 2009.
Some analysis shows that the only sets of 4 digit squares with no repeat digits such that 4 or more share the same sod are:
18: [1089, 1296, 1764, 2916, 3249, 4356, 4761, 5184, 6084, 7056, 9216, 9801]
19: [1369, 1936, 2809, 4096, 5329, 6724, 7921, 9604]
(first step was to make dictionary I called samesod)
From the sod() = 18 group, the only set of 4 which has different digits in each column (condition 3) is:
[Edit, correction: the only set of 4 which have the same sum of digits for each column]
(1764, 3249, 5184, 9801)
From the sod() = 19 group, no subset of 4 met condition 3.
Solution (with the rows in any order):
1764
3249
5184
9801
-----------------
Dictionary "samesod" printed out
{7: [1024, 2401],
18: [1089, 1296, 1764, 2916, 3249, 4356, 4761, 5184, 6084, 7056, 9216, 9801],
19: [1369, 1936, 2809, 4096, 5329, 6724, 7921, 9604],
22: [1849, 5476],
9: [2304, 2601],
13: [2704, 3721, 6241],
10: [3025, 5041],
16: [3481, 9025],
25: [7396],
27: [7569, 8649]}
The results for 18 and 19 'hardwired' into the program for the next step
def sod(n):
""" Input an integer. Returns the Sum of the Digits """
aList = list(str(n))
ans = 0
for c in aList:
ans = ans + int(c)
return ans
squares = [i**2 for i in range(32,100) if len(set(str(i**2))) == 4]
samesod = {}
for s in squares:
sods = sod(s)
if sods not in samesod:
samesod[sods] = [s]
else:
samesod[sods].append(s)
sod18 = [1089, 1296, 1764, 2916, 3249, 4356, 4761, 5184, 6084, 7056, 9216, 9801]
sod19 = [1369, 1936, 2809, 4096, 5329, 6724, 7921, 9604]
grids = []
from itertools import combinations
for c in combinations(sod18,4):
#for c in combinations(sod19,4): #run either 18 or 19
grid = [[d for d in str(c[i])] for i in range(4)]
reject = False
flip = [[grid[j][i] for j in range(4) ]for i in range(4)]
columnsods = [ sod( int(''.join(f)) ) for f in flip]
if len(set(columnsods)) != 1:
reject = True
if reject:
continue
grids.append(c)
print(grids)
Edited on July 21, 2023, 9:48 am
Edited on July 21, 2023, 11:05 am
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Posted by Larry
on 2023-07-21 09:46:04 |
Solution
