Puzzle for August 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
Add C to both sides of eq.2: C – E + C = A – C + C which becomes eq.2a) 2×C – E = A Add A and B to both sides of eq.4: D – A + A + B = A – B – C + A + B which becomes eq.4a) D + B = 2×A – C
Hint #2
In eq.4a, substitute (2×C – E) for A (from eq.2a): D + B = 2×(2×C – E) – C which becomes D + B = 4×C – 2×E – C which becomes D + B = 3×C – 2×E which is the same as eq.4b) B + D = 3×C – 2×E
Hint #3
Add C and D to both sides of eq.5: B – C + C + D = C – D + E + C + D which becomes eq.5a) B + D = 2×C + E
Hint #4
In eq.5a, replace B + D with 3×C – 2×E (from eq.4b): 3×C – 2×E = 2×C + E In the above equation, add 2×E to both sides, and subtract 2×C from both sides: 3×C – 2×E + 2×E – 2×C = 2×C + E + 2×E – 2×C which simplifies to C = 3×E
Hint #5
In eq.2a, replace C with (3×E): 2×(3×E) – E = A which becomes 6×E – E = A which makes 5×E = A
Hint #6
In eq.4b, replace C with (3×E): B + D = 3×(3×E) – 2×E which becomes B + D = 9×E – 2×E which becomes eq.4c) B + D = 7×E
Hint #7
Add A and E to both sides of eq.6: A + C + E + A + E = B + D + F – A – E + A + E which becomes 2×A + C + 2×E = B + D + F In the above equation, substitute (5×E) for A, 3×E for C, and 7×E for B + D (from eq.4c): 2×(5×E) + 3×E + 2×E = 7×E + F which becomes 10×E + 5×E = 7×E + F which becomes 15×E = 7×E + F Subtract 7×E from each side: 15×E – 7×E = 7×E + F – 7×E which becomes 8×E = F
Hint #8
Substitute 5×E for A, 3×E for C, and 8×E for F in eq.3: 5×E + D = 3×E + E + 8×E which becomes 5×E + D = 12×E Subtract 5×E from each side of the equation above: 5×E + D – 5×E = 12×E – 5×E which becomes D = 7×E
Hint #9
Substitute 7×E for D in eq.4c: B + 7×E = 7×E Subtract 7×E from each side of the equation above: B + 7×E – 7×E = 7×E – 7×E which makes B = 0
Solution
Substitute 5×E for A, 0 for B, 3×E for C, 7×E for D, and 8×E for F in eq.1: 5×E + 0 + 3×E + 7×E + E + 8×E = 24 which simplifies to 24×E = 24 Divide both sides of the above equation by 24: 24×E ÷ 24 = 24 ÷ 24 which means E = 1 making A = 5×E = 5 × 1 = 5 C = 3×E = 3 × 1 = 3 D = 7×E = 7 × 1 = 7 F = 8×E = 8 × 1 = 8 and ABCDEF = 503718