Puzzle for June 25, 2020 ( )
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.5, replace D with C + F (from eq.4): B + C + C + F = E + F which becomes B + 2×C + F = E + F Subtract F from both sides of the above equation: B + 2×C + F – F = E + F – F which becomes eq.5a) B + 2×C = E
Hint #2
In eq.5a, replace E with B + D (from eq.3): B + 2×C = B + D Subtract B from each side of the above equation: B + 2×C – B = B + D – B which makes 2×C = D
Hint #3
In eq.4, substitute 2×C for D: C + F = 2×C Subtract C from each side of the equation above: C + F – C = 2×C – C which makes F = C
Hint #4
Substitute 2×C for D in eq.6: A + 2×C – B = B + C + E In the equation above, add B to both sides, and subtract C from both sides: A + 2×C – B + B – C = B + C + E + B – C which becomes eq.6a) A + C = 2×B + E
Hint #5
Substitute B + 2×C for E (from eq.5a) in eq.6a: A + C = 2×B + B + 2×C which becomes A + C = 3×B + 2×C Subtract C from both sides of the above equation: A + C – C = 3×B + 2×C – C which becomes eq.6b) A = 3×B + C
Hint #6
Substitute 2×C for D, and 3×B + C for A (from eq.6b) in eq.2: 2×C = 3×B + C + B which becomes 2×C = 4×B + C Subtract C from both sides of the equation above: 2×C – C = 4×B + C – C which makes C = 4×B and also makes F = C = 4×B
Hint #7
Substitute 4×B for C in eq.6b: A = 3×B + 4×B which makes A = 7×B
Hint #8
Substitute 4×B for C and F in eq.4: 4×B + 4×B = D which makes 8×B = D
Hint #9
Substitute (4×B) for C in eq.5a: B + 2×(4×B) = E which is equivalent to B + 8×B = E which makes 9×B = E
Solution
Substitute 7×B for A, 4×B for C and F, 8×B for D, and 9×B for E in eq.1: 7×B + B + 4×B + 8×B + 9×B + 4×B = 33 which simplifies to 33×B = 33 Divide both sides of the above equation by 33: 33×B ÷ 33 = 33 ÷ 33 which means B = 1 making A = 7×B = 7 × 1 = 7 C = F = 4×B = 4 × 1 = 4 D = 8×B = 8 × 1 = 8 E = 9×B = 9 × 1 = 9 and ABCDEF = 714894