Puzzle for January 6, 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
eq.6 may be written as: E = A + C + F – E In the above equation, replace C + F with E (from eq.5): E = A + E – E which makes E = A
Hint #2
In eq.4, replace E with A: B + A = A – B + C In the equation above, subtract A from both sides, and add B to both sides: B + A – A + B = A – B + C – A + B which makes 2×B = C
Hint #3
In eq.2, substitute A – C – D + E for B + C + D (from eq.3): eq.2a) F = A – C – D + E – F
Hint #4
In eq.2a, substitute C + F for E (from eq.5), and E for A: F = E – C – D + C + F – F which becomes eq.2b) F = E – D
Hint #5
Substitute E – D for F (from eq.2b) in eq.5: C + E – D = E In the above equation, subtract E from both sides, and add D to both sides: C + E – D – E + D = E – E + D which makes C = D and also makes C = D = 2×B
Hint #6
In eq.3, substitute 2×B for C and D, and A for E: A – 2×B – 2×B + A = B + 2×B + 2×B which becomes 2×A – 4×B = 5×B Add 4×B to both sides of the above equation: 2×A – 4×B + 4×B = 5×B + 4×B which becomes 2×A = 9×B Divide both sides by 2: 2×A ÷ 2 = 9×B ÷ 2 which makes A = 4½×B and also makes E = A = 4½×B
Hint #7
Substitute 4½×B for E, and 2×B for D in eq.2b: F = 4½×B – 2×B which makes F = 2½×B
Solution
Substitute 4½×B for A and E, 2×B for C and D, and 2½×B for F in eq.1: 4½×B + B + 2×B + 2×B + 4½×B + 2½×B = 33 which simplifies to 16½×B = 33 Divide both sides of the above equation by 16½: 16½×B ÷ 16½ = 33 ÷ 16½ which means B = 2 making A = E = 4½×B = 4½ × 2 = 9 C = D = 2×B = 2 × 2 = 4 F = 2½×B = 2½ × 2 = 5 and ABCDEF = 924495