Puzzle for July 20, 2021 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
Scratchpad
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Hint #1
In eq.5, replace D with E + F (from eq.3): E + F + E = A + F which becomes 2×E + F = A + F Subtract F from each side of the above equation: 2×E + F – F = A + F – F which makes 2×E = A
Hint #2
In eq.6, replace B with D + F (from eq.2), and C with A + D (from eq.4): D + F + F = A + D + D + E which becomes D + 2×F = A + 2×D + E Subtract D from both sides of the above equation: D + 2×F – D = A + 2×D + E – D which becomes eq.6a) 2×F = A + D + E
Hint #3
In eq.6a, substitute 2×E for A, and E + F for D (from eq.3): 2×F = 2×E + E + F + E which becomes 2×F = 4×E + F Subtract F from both sides of the equation above: 2×F – F = 4×E + F – F which makes F = 4×E
Hint #4
Substitute 2×E for A, and 4×E for F in eq.5: D + E = 2×E + 4×E which becomes D + E = 6×E Subtract E from each side of the equation above: D + E – E = 6×E – E which makes D = 5×E
Hint #5
Substitute 2×E for A, and 5×E for D in eq.4: C = 2×E + 5×E which makes C = 7×E
Hint #6
Substitute 5×E for D, and 4×E for F in eq.2: 5×E + 4×E = B which makes 9×E = B
Solution
Substitute 2×E for A, 9×E for B, 7×E for C, 5×E for D, and 4×E for F in eq.1: 2×E + 9×E + 7×E + 5×E + E + 4×E = 28 which simplifies to 28×E = 28 Divide both sides of the equation above by 28: 28×E ÷ 28 = 28 ÷ 28 which means E = 1 making A = 2×E = 2 × 1 = 2 B = 9×E = 9 × 1 = 9 C = 7×E = 7 × 1 = 7 D = 5×E = 5 × 1 = 5 F = 4×E = 4 × 1 = 4 and ABCDEF = 297514