Another of Elsa's Smallest Tile
Tiles = { 0 , 1 , 2 , 3 , 4 , 5 }
Dora : The sum of squares of my tiles equals the sum of Elsa's tiles.
Elsa : We all have exactly two different tiles.
Farrah : The sum of squares of my tiles is as big as Elsa's largest tile.
What is Elsa's smallest tile?
The answer is 4.
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1 solution
Let Dora have tiles a & b, Elsa's c & d and Farrah's e & f, with a < b, c < d and e < f wlog. Quite obviously from Dora's statement simplified to a² + b² = c + d, Dora's and Elsa's together make even numbers of even and odd tiles. Since we have 3 each altogether, theirs must be 2 evens and 2 odds, leaving Farrah to take the rest (1 odd + 1 even). From #3, e² + f² = d ≤ 5 ---> f ≤ 2.
(e,f) = {0,1,2}, where 1 is the only odd number possible for Farrah's odd tile. From the remaining 2 odds, only d = 5 have a square difference with Farrah's odd square. So (d,e,f) = (5,1,2).
From #1, a² + b² = c + d = c + 5. Considering that the 2 largest available tiles add up to less than 10, therefore a and b can't be too big and thus, 4 must be c and Dora's tiles are 0 & 3.
Answer = 4