Elsa's Forever and Always

Logic Level 3

Tiles = { 0 , 1 , 2 , 3 , 4 , 5 }

Dora : The difference of squares of my tiles is half the difference of squares of Elsa's tiles.

Elsa : We all have exactly two different tiles.

Farrah : The difference of squares of one of Dora's and one of Elsa's tiles is as big as the difference of my tiles.

What will always be Elsa's tile?

0 1 2 3 4 5

2 solutions

A Former Brilliant Member
Feb 4, 2021

from itertools import permutations
# Dora tiles: d1, d2
# Elsa tiles: e1, e2
# Farrah tiles: f1, f2
for d1, d2, e1, e2, f1, f2 in permutations('012345',6):
  if int(d1)**2-int(d2)**2 == 1/2*(int(e1)**2-int(e2)**2) and int(d1)**2-int(e1)**2 == int(f1)-int(f2):
    print(int(e1), int(e2))

1
2
3

output:
1 3
1 5

Saya Suka
Jun 9, 2021

Let's assign the letters D and d as Dora's numbers, E and e as Elsa's and finally F and f as Farrah's, with capital letters as the bigger numbers.

D² – d² = (E² – e²) / 2
2(D – d)(D + d) = (E – e)(E + e)

Because RHS is divisible by 2, Elsa's numbers must have the same parity, which will make her difference of squares a multiple of 4 (at least 2 factors of 2) and consequently, the LHS of Dora's difference of squares also a multiple of 2 and her numbers of the same parity, too. Since the available numbers only consist of 3 odd and 3 even digits, taking out two pairs of same parity numbers will leave a pair of opposite parity numbers for Farrah. As such, the difference of squares of one of Dora's and one of Elsa's must be an odd number less than 6 (maximized with 5 (max) minus 0 (min) equal to 5), so that'll be 1 or 3 or 5.

Possibilities :
a) 5 = 5 – 0 = 3² – 2²
Farrah's = (5, 0), the others are (3, 1) & (4, 2).

b) 3 = 2² – 1² = 3 – 0
Farrah's = (3, 0), the others are (5, 1) & (4, 2).

c) 1 = 1² – 0² = 5 – 4 = 4 – 3 = 3 – 2
ci) Farrah's = (5, 4), the others are (3, 1) & (2, 0).
cii) Farrah's = (4, 3), the others are (5, 1) & (2, 0).
ciii) Farrah's = (3, 2), the others are (5, 1) & (4, 0).

Even against the truth that some of the above 5 possibilities are not a solution (according to the 3 hints by the girls), we can already see that none of Farrah's can include a 1, meaning that 1 is always one of the odd pair. To get the real answer, testing all the possibilities whether they are in compliance with Dora's clue leave us with just possibilities b) and ci) where Elsa always have the odd numbers and Dora's the even ones.

Elsa's Forever and Always

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