Puzzle for March 26, 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 A and C to both sides of eq.6: F – A + A + C = A – C + A + C which becomes eq.6a) F + C = 2×A In eq.3, replace B + E with A (from eq.2): C + D = A + A which becomes eq.3a) C + D = 2×A
Hint #2
In eq.6a, replace 2×A with C + D (from eq.3a): F + C = C + D Subtract C from both sides of the above equation: F + C – C = C + D – C which makes F = D
Hint #3
In eq.3, substitute F for D, and B + F for E (from eq.5): C + F = A + B + B + F which becomes C + F = A + 2×B + F Subtract F and 2×B from each side of the above equation: C + F – F – 2×B = A + 2×B + F – F – 2×B which becomes eq.3b) C – 2×B = A
Hint #4
Substitute (C – 2×B) for A (from eq.3b) in eq.3a: C + D = 2×(C – 2×B) which becomes C + D = 2×C – 4×B In the above equation, subtract C from both sides, and add 4×B to both sides: C + D – C + 4×B = 2×C – 4×B – C + 4×B which becomes eq.3c) D + 4×B = C
Hint #5
Substitute D for F, and D + 4×B for C (from eq.3c) in eq.4: D + D = B + D + 4×B which becomes 2×D = 5×B + D Subtract D from both sides of the above equation: 2×D – D = 5×B + D – D which makes D = 5×B and also makes F = D = 5×B
Hint #6
Substitute 5×B for D in eq.3c: 5×B + 4×B = C which makes 9×B = C
Hint #7
Substitute 9×B for C in eq.3b: 9×B – 2×B = A which makes 7×B = A
Hint #8
Substitute 5×B for F in eq.5: E = B + 5×B which makes E = 6×B
Solution
Substitute 7×B for A, 9×B for C, 5×B for D and F, and 6×B for E in eq.1: 7×B + B + 9×B + 5×B + 6×B + 5×B = 33 which simplifies to 33×B = 33 Divide both sides of the equation above by 33: 33×B ÷ 33 = 33 ÷ 33 which means B = 1 making A = 7×B = 7 × 1 = 7 C = 9×B = 9 × 1 = 9 D = F = 5×B = 5 × 1 = 5 E = 6×B = 6 × 1 = 6 and ABCDEF = 719565