Puzzle for June 15, 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.2, substitute (B + C) for D (from eq.4): E – (B + C) = B – C which is equivalent to E – B – C = B – C Add B and C to each side of the equation above: E – B – C + B + C = B – C + B + C which simplifies to E = 2×B
Hint #2
In eq.6, replace A with C + F (from eq.3), and replace E with 2×B: F – B = C + F – 2×B In the equation above, subtract F from both sides, and add 2×B to both sides: F – B – F + 2×B = C + F – 2×B – F + 2×B which simplifies to B = C
Hint #3
In eq.4, replace C with B: D = B + B which makes D = 2×B
Hint #4
In eq.5, substitute B for C, and 2×B for D and E: B + B + 2×B + 2×B = F – 2×B which becomes 6×B = F – 2×B Add 2×B to both sides of the above equation: 6×B + 2×B = F – 2×B + 2×B which makes 8×B = F
Hint #5
Substitute B for C, and 8×B for F in eq.3: B + 8×B = A which makes 9×B = A
Solution
Substitute 9×B for A, B for C, 2×B for D and E, and 8×B for F in eq.1: 9×B + B + B + 2×B + 2×B + 8×B = 23 which simplifies to 23×B = 23 Divide both sides of the equation above by 23: 23×B ÷ 23 = 23 ÷ 23 which means B = 1 making A = 9×B = 9 × 1 = 9 C = B = 1 D = E = 2×B = 2 × 1 = 2 F = 8×B = 8 × 1 = 8 and ABCDEF = 811922