Puzzle for April 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 E to both sides of eq.5: D – E + E = A + E + E which becomes eq.5a) D = A + 2×E In eq.4, replace D with A + 2×E (from eq.5a): A + C = A + 2×E + E which becomes A + C = A + 3×E Subtract A from each side of the above equation: A + C – A = A + 3×E – A which makes C = 3×E
Hint #2
In eq.2, replace C with 3×E: F = 3×E + E which makes F = 4×E
Hint #3
Add A to both sides of eq.6: B – A + A = A + E + F + A which becomes eq.6a) B = 2×A + E + F In eq.6a, substitute B – D for E + F (from eq.3): B = 2×A + B – D In the equation above, subtract B from both sides, and add D to both sides: B – B + D = 2×A + B – D – B + D which simplifies to eq.3a) D = 2×A
Hint #4
Sustitute 2×A for D (from eq.3a) into eq.5a: 2×A = A + 2×E Subtract A from each side of the equation above: 2×A – A = A + 2×E – A which makes A = 2×E
Hint #5
Sustitute (2×E) for A in eq.3a: D = 2×(2×E) which becomes D = 4×E
Hint #6
Substitute (2×E) for A, and 4×E for F in eq.6a: B = 2×(2×E) + E + 4×E which becomes B = 4×E + 5×E which makes B = 9×E
Solution
Substitute 2×E for A, 9×E for B, 3×E for C, and 4×E for D and F in eq.1: 2×E + 9×E + 3×E + 4×E + E + 4×E = 23 which simplifies to 23×E = 23 Divide both sides of the above equation by 23: 23×E ÷ 23 = 23 ÷ 23 which means E = 1 making A = 2×E = 2 × 1 = 2 B = 9×E = 9 × 1 = 9 C = 3×E = 3 × 1 = 3 D = F = 4×E = 4 × 1 = 4 and ABCDEF = 293414