Puzzle for August 1, 2019 ( )
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.2, replace D with B + E (from eq.4): B + E = A + B + C Subtract B from each side of the above equation: B + E – B = A + B + C – B which becomes eq.2a) E = A + C
Hint #2
In eq.5, replace E with A + C (from eq.2a): C = A + C – F In the above equation, subtract C from each side, and add F to each side: C – C + F = A + C – F – C + F which simplifies to F = A
Hint #3
In eq.3, replace F with A: E + A = A + B Subtract A from each side of the above equation: E + A – A = A + B – A which makes E = B
Hint #4
In eq.4, substitute E for B: E + E = D which means eq.4a) 2×E = D
Hint #5
Add C to both sides of eq.6: A + B – C + F + C = C + D + C which becomes A + B + F = 2×C + D Substitute A for F, E for B, and 2×E for D in the above equation: A + E + A = 2×C + 2×E which becomes 2×A + E = 2×C + 2×E Subtract E from both sides: 2×A + E – E = 2×C + 2×E – E which becomes eq.6a) 2×A = 2×C + E
Hint #6
Substitute A + C for E (from eq.2a) in eq.6a: 2×A = 2×C + A + C which becomes 2×A = 3×C + A Subtract A from both sides: 2×A – A = 3×C + A – A which becomes A = 3×C and which also makes F = A = 3×C
Hint #7
Substitute 3×C for A in eq.2a: E = 3×C + C which makes E = 4×C and also makes B = E = 4×C
Hint #8
Substitute 4×C for E in eq.4a: 2×(4×C) = D which means 8×C = D
Solution
Substitute 3×C for A and F, 4×C for B and E, and 8×C for D in eq.1: 3×C + 4×C + C + 8×C + 4×C + 3×C = 23 which simplifies to 23×C = 23 Divide both sides by 23: 23×C ÷ 23 = 23 ÷ 23 which means C = 1 making A = F = 3×C = 3 × 1 = 3 B = E = 4×C = 4 × 1 = 4 D = 8×C = 8 × 1 = 8 and ABCDEF = 341843