Puzzle for August 5, 2019 ( )
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, replace C with A + F (from eq.4), and D with A + E (from eq.3): B + A + F = A + E + F Subtract both A and F from each side of the above equation: B + A + F – A – F = A + E + F – A – F which simplifies to B = E
Hint #2
In eq.5, replace E with B: F = B + B which becomes F = 2×B
Hint #3
In eq.6, substitute B for E, and 2×B for F: A + B = B + 2×B which becomes A + B = 3×B Subtract B from each side of the above equation: A + B – B = 3×B – B which makes A = 2×B
Hint #4
Substitute 2×B for both A and F in eq.4: C = 2×B + 2×B which makes C = 4×B
Hint #5
Substitute 2×B for A, and B for E in eq.3: D = 2×B + B which makes D = 3×B
Solution
Substitute 2×B for A and F, 4×B for C, 3×B for D, and B for E in eq.1: 2×B + B + 4×B + 3×B + B + 2×B = 26 which simplifies to 13×B = 26 Divide both sides of the equation above by 13: 13×B ÷ 13 = 26 ÷ 13 which means B = 2 making A = F = 2×B = 2 × 2 = 4 C = 4×B = 4 × 2 = 8 D = 3×B = 3 × 2 = 6 E = B = 2 and ABCDEF = 428624