Puzzle for June 9, 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.4, replace B + D with E + F (from eq.3): C + E = E + F + F which becomes C + E = E + 2×F Subtract E from each side of the equation above: C + E – E = E + 2×F – E which makes C = 2×F
Hint #2
In eq.2, replace C with 2×F: D = 2×F + F which makes D = 3×F
Hint #3
In eq.5, substitute 3×F for D: 3×F – A = A – F Add A and F to both sides of the above equation: 3×F – A + A + F = A – F + A + F which becomes 4×F = 2×A Divide both sides by 2: 4×F ÷ 2 = 2×A ÷ 2 which makes 2×F = A
Hint #4
Substitute 2×F for A and C, and 3×F for D in eq.6: 2×F – B + 2×F = B + 3×F + F – (2×F – B) which becomes 4×F – B = B + 4×F – 2×F + B which becomes 4×F – B = 2×B + 2×F In the equation above, add B to each side, and subtract 2×F from each side: 4×F – B + B – 2×F = 2×B + 2×F + B – 2×F which makes 2×F = 3×B Divide both sides by 3: 2×F ÷ 3 = 3×B ÷ 3 which makes ⅔×F = B
Hint #5
Substitute ⅔×F for B, and 3×F for D in eq.3: ⅔×F + 3×F = E + F which becomes 3⅔×F = E + F Subtract F from each side of the equation above: 3⅔×F – F = E + F – F which makes 2⅔×F = E
Solution
Substitute 2×F for A and C, ⅔×F for B, 3×F for D, and 2⅔×F for E in eq.1: 2×F + ⅔×F + 2×F + 3×F + 2⅔×F + F = 34 which simplifies to 11⅓×F = 34 Divide both sides of the equation above by 11⅓: 11⅓×F ÷ 11⅓ = 34 ÷ 11⅓ which means F = 3 making A = C = 2×F = 2 × 3 = 6 B = ⅔×F = ⅔ × 3 = 2 D = 3×F = 3 × 3 = 9 E = 2⅔×F = 2⅔ × 3 = 8 and ABCDEF = 626983