Puzzle for December 21, 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
In eq.3, replace F with A + C (from eq.2): C + A + C = A + E which becomes 2×C + A = A + E Subtract A from each side of the above equation: 2×C + A – A = A + E – A which makes 2×C = E
Hint #2
Subtract E from each side of eq.5: C + D – E = A + B + E – E which becomes C + D – E = A + B In eq.6, replace A + B with C + D – E: D + E = C + D – E + F In the above equation, subtract D from both sides, and add E to both sides: D + E – D + E = C + D – E + F – D + E which simplifies to eq.6a) 2×E = C + F
Hint #3
In eq.6a, substitute (2×C) for E: 2×(2×C) = C + F which becomes 4×C = C + F Subtract C from both sides of the equation above: 4×C – C = C + F – C which makes 3×C = F
Hint #4
Substitute 3×C for F in eq.2: 3×C = A + C Subtract C from both sides of the above equation: 3×C – C = A + C – C which makes 2×C = A
Hint #5
Substitute 2×C for E, and 3×C for F in eq.4: 2×C + 3×C = D – 3×C which becomes 5×C = D – 3×C Add 3×C to both sides of the above equation: 5×C + 3×C = D – 3×C + 3×C which makes 8×C = D
Hint #6
Substitute 8×C for D, and 2×C for A and E in eq.5: C + 8×C = 2×C + B + 2×C which becomes 9×C = 4×C + B Subtract 4×C from each side of the above equation: 9×C – 4×C = 4×C + B – 4×C which makes 5×C = B
Solution
Substitute 2×C for A and E, 5×C for B, 8×C for D, and 3×C for F in eq.1: 2×C + 5×C + C + 8×C + 2×C + 3×C = 21 which simplifies to 21×C = 21 Divide both sides of the above equation by 21: 21×C ÷ 21 = 21 ÷ 21 which means C = 1 making A = E = 2×C = 2 × 1 = 2 B = 5×C = 5 × 1 = 5 D = 8×C = 8 × 1 = 8 F = 3×C = 3 × 1 = 3 and ABCDEF = 251823