Puzzle for January 14, 2020 ( )
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.4, replace F with C + D (from eq.3), and replace E with A + B + C (from eq.2): B + C + D = A + A + B + C Subtract B and C from each side of the equation above: B + C + D – B – C = A + A + B + C – B – C which simplifies to D = 2×A
Hint #2
In eq.5, subtract A from each side, and add C to each side: D – A + C – A + C = A – C – A + C which becomes D – 2×A + 2×C = 0 In the equation above, replace 2×A with D: D – D + 2×C = 0 which makes 2×C = 0 which means C = 0
Hint #3
In eq.3, substitute 0 for C: 0 + D = F which makes D = F and which also makes eq.3a) D = F = 2×A
Hint #4
Substitute 0 for C in eq.2: E = A + B + 0 eq.2a) E = A + B
Hint #5
In eq.6, substitute 2×A for D, and (A + B) for E (from eq.2a): A – B + 2×A – (A + B) = B + (A + B) which becomes 3×A – B – A – B = A + 2×B which becomes 2×A – 2×B = A + 2×B In the equation above, add 2×B to each side, and subtract A from both sides: 2×A – 2×B + 2×B – A = A + 2×B + 2×B – A which makes A = 4×B
Hint #6
Substitute 4×B for A in eq.2a: E = 4×B + B which makes E = 5×B
Hint #7
Substitute (4×B) for A in eq.3a: D = F = 2×(4×B) which makes D = F = 8×B
Solution
Substitute 4×B for A, 0 for C, 8×B for D and F, and 5×B for E in eq.1: 4×B + B + 0 + 8×B + 5×B + 8×B = 26 which simplifies to 26×B = 26 Divide both sides of the equation above by 26: 26×B ÷ 26 = 26 ÷ 26 which means B = 1 making A = 4×B = 4×1 = 4 D = F = 8×A = 8×1 = 8 E = 5×B = 5×1 = 5 and ABCDEF = 410858