Puzzle for May 5, 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
Add B to both sides of eq.2: D – B + B = A + B which becomes eq.2a) D = A + B Add E to both sides of eq.5: D – E + E = A + E + E which becomes eq.5a) D = A + 2×E
Hint #2
In eq.5a, replace D with A + B (from eq.2a): A + B = A + 2×E Subtract A from each side of the above equation: A + B – A = A + 2×E – A which makes eq.5b) B = 2×E
Hint #3
In eq.4, replace B with 2×E: E – F = 2×E – E + F which becomes E – F = E + F In the above equation, subtract E from both sides, and add F to both sides: E – F – E + F = E + F – E + F which makes 0 = 2×F which means 0 = F
Hint #4
In eq.6, substitute A + 2×E for D (from eq.5a), and 2×E for B: C + A + 2×E – A – 2×E = A + 2×E – E which becomes eq.6a) C = A + E
Hint #5
In eq.3, substitute 2×E for B, and (A + E) for C (from eq.6a): 2×E + E – A – (A + E) = A + (A + E) which becomes 2×E + E – A – A – E = 2×A + E which becomes 2×E – 2×A = 2×A + E In the above equation, add 2×A to both sides, and subtract E from both sides: 2×E – 2×A + 2×A – E = 2×A + E + 2×A – E which makes E = 4×A
Hint #6
Substitute (4×A) for E in eq.5b: B = 2×(4×A) which makes B = 8×A
Hint #7
Substitute 4×A for E in eq.6a: C = A + 4×A which makes C = 5×A
Hint #8
Substitute (4×A) for E in eq.5a: D = A + 2×(4×A) which becomes D = A + 8×A which makes D = 9×A
Solution
Substitute 8×A for B, 5×A for C, 9×A for D, 4×A for E, and 0 for F in eq.1: A + 8×A + 5×A + 9×A + 4×A + 0 = 27 which simplifies to 27×A = 27 Divide both sides of the above equation by 27: 27×A ÷ 27 = 27 ÷ 27 which means A = 1 making B = 8×A = 8 × 1 = 8 C = 5×A = 5 × 1 = 5 D = 9×A = 9 × 1 = 9 E = 4×A = 4 × 1 = 4 and ABCDEF = 185940