Puzzle for December 10, 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 F to both sides of eq.3: E – F + F = B – D + F + F which becomes E = B – D + 2×F In eq.2, replace E with B – D + 2×F: B + B – D + 2×F = A + C which becomes eq.2a) 2×B – D + 2×F = A + C
Hint #2
In eq.6, add B to both sides, and subtract D from both sides: B + C + F + B – D = A – B + D + B – D which becomes 2×B + C + F – D = A In eq.2a, replace A with 2×B + C + F – D: 2×B – D + 2×F = 2×B + C + F – D + C which becomes 2×B – D + 2×F = 2×B + 2×C + F – D In the equation above, subtract 2×B and F from both sides, and add D to both sides: 2×B – D + 2×F – 2×B – F + D = 2×B + 2×C + F – D – 2×B – F + D which simplifies to F = 2×C
Hint #3
Add C to both sides of eq.5: C + E + C = A – C + D + C which becomes 2×C + E = A + D In the above equation, substitute F for 2×C: F + E = A + D which may be written as eq.5a) E + F = A + D
Hint #4
Substitute A + D for E + F (from eq.5a) into eq.4: A + D = A + D – D which becomes A + D = A Subtract A from each side of the equation above: A + D – A = A – A which makes D = 0
Hint #5
Substitute 0 for D in eq.5a: E + F = A + 0 which becomes eq.5b) E + F = A
Hint #6
Substitute E + F for A (from eq.5b) into eq.2: B + E = E + F + C Subtract E from each side of the equation above: B + E – E = E + F + C – E which becomes eq.2b) B = F + C
Hint #7
Substitute 2×C for F in eq.2b: B = 2×C + C which makes B = 3×C
Hint #8
Substitute 2×C for F, 3×C for B, and 0 for D in eq.3: E – 2×C = 3×C – 0 + 2×C which becomes E – 2×C = 5×C Add 2×C to both sides of the equation above: E – 2×C + 2×C = 5×C + 2×C which makes E = 7×C
Hint #9
Substitute 7×C for E, and 2×C for F in eq.5b: 7×C + 2×C = A which makes 9×C = A
Solution
Substitute 9×C for A, 3×C for B, 0 for D, 7×C for E, and 2×C for F in eq.1: 9×C + 3×C + C + 0 + 7×C + 2×C = 22 which simplifies to 22×C = 22 Divide both sides of the above equation by 22: 22×C ÷ 22 = 22 ÷ 22 which means C = 1 making A = 9×C = 9 × 1 = 9 B = 3×C = 3 × 1 = 3 E = 7×C = 7 × 1 = 7 F = 2×C = 2 × 1 = 2 and ABCDEF = 931072