Puzzle for June 27, 2022 ( )
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.6, replace E with A + B (from eq.2): B + A + B = A + D which becomes 2×B + A = A + D Subtract A from each side of the above equation: 2×B + A – A = A + D – A which makes 2×B = D
Hint #2
In eq.3, replace D with 2×B: 2×B = B + C Subtract B from each side of the equation above: 2×B – B = B + C – B which makes B = C
Hint #3
In eq.4, substitute B for C, and 2×B for D: F = B + 2×B which makes F = 3×B
Hint #4
Substitute 2×B for D, and 3×B for F in eq.5: 2×B + 3×B = A – 3×B which becomes 5×B = A – 3×B Add 3×B to both sides of the above equation: 5×B + 3×B = A – 3×B + 3×B which makes 8×B = A
Hint #5
Substitute 8×B for A in eq.2: E = 8×B + B which makes E = 9×B
Solution
Substitute 8×B for A, B for C, 2×B for D, 9×B for E, and 3×B for F in eq.1: 8×B + B + B + 2×B + 9×B + 3×B = 24 which simplifies to 24×B = 24 Divide both sides of the above equation by 24: 24×B ÷ 24 = 24 ÷ 24 which means B = 1 making A = 8×B = 8 × 1 = 8 C = B = 1 D = 2×B = 2 × 1 = 2 E = 9×B = 9 × 1 = 9 F = 3×B = 3 × 1 = 3 and ABCDEF = 811293