Puzzle for July 12, 2022 ( )
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 A with D + E (from eq.4): C + D = D + E + E which becomes C + D = D + 2×E Subtract D from each side of the equation above: C + D – D = D + 2×E – D which makes C = 2×E
Hint #2
In eq.6, replace B with C + F (from eq.3): C + F + F = C + A which becomes C + 2×F = C + A Subtract C from each side of the above equation: C + 2×F – C = C + A – C which makes eq.6a) 2×F = A
Hint #3
In eq.6, substitute B + C for A (from eq.2): B + F = B + C + C which becomes B + F = B + 2×C Subtract B from both sides of the equation above: B + F – B = B + 2×C – B which makes eq.6b) F = 2×C
Hint #4
Substitute (2×E) for C in eq.6b: F = 2×(2×E) which makes F = 4×E
Hint #5
Substitute (4×E) for F in eq.6a: 2×(4×E) = A which makes 8×E = A
Hint #6
Substitute 2×E for C, and 4×E for F in eq.3: B = 2×E + 4×E which makes B = 6×E
Hint #7
Substitute 8×E for A in eq.4: 8×E = D + E Subtract E from each side of the equation above: 8×E – E = D + E – E which makes 7×E = D
Solution
Substitute 8×E for A, 6×E for B, 2×E for C, 7×E for D, and 4×E for F in eq.1: 8×E + 6×E + 2×E + 7×E + E + 4×E = 28 which simplifies to 28×E = 28 Divide both sides of the above equation by 28: 28×E ÷ 28 = 28 ÷ 28 which means E = 1 making A = 8×E = 8 × 1 = 8 B = 6×E = 6 × 1 = 6 C = 2×E = 2 × 1 = 2 D = 7×E = 7 × 1 = 7 F = 4×E = 4 × 1 = 4 and ABCDEF = 862714