Puzzle for April 8, 2021 ( )
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.
Our thanks to Abby S (age 10) for sending us this puzzle! Thank you, Abby!
Scratchpad
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Hint #1
In eq.6, replace D + E with B (from eq.2): B + C – D = B + F In the above equation, subtract B from both sides, and add D to both sides: B + C – D – B + D = B + F – B + D which becomes eq.6a) C = F + D
Hint #2
In eq.3, replace C with F + D (from eq.6a): D + E = F + D + F which becomes D + E = 2×F + D Subtract D from each side of the equation above: D + E – D = 2×F + D – D which makes E = 2×F
Hint #3
In eq.5, substitute 2×F for E, and F + D for C (from eq.6a): A + 2×F = F + D + D – 2×F which becomes A + 2×F = 2×D – F Subtract 2×F from each side of the equation above: A + 2×F – 2×F = 2×D – F – 2×F which becomes eq.5a) A = 2×D – 3×F
Hint #4
Substitute B for D + E (from eq.2), and F + D for C (from eq.6a) into eq.3: B = F + D + F which becomes eq.3a) B = D + 2×F
Hint #5
In eq.4, substitute (F + D) for C (from eq.6a), 2×F for E, 2×D – 3×F for A (from eq.5a), and D + 2×F for B (from eq.3a): (F + D) + 2×F = 2×D – 3×F + D + 2×F – (F + D) which becomes D + 3×F = 3×D – F – F – D which becomes D + 3×F = 2×D – 2×F In the above equation, add 2×F to both sides, and subtract D from both sides: D + 3×F + 2×F – D = 2×D – 2×F + 2×F – D which makes 5×F = D
Hint #6
Substitute (5×F) for D in eq.5a: A = 2×(5×F) – 3×F which becomes A = 10×F – 3×F which makes A = 7×F
Hint #7
Substitute 5×F for D in eq.3a: B = 5×F + 2×F which makes B = 7×F
Hint #8
Substitute 5×F for D in eq.6a: C = F + 5×F which makes C = 6×F
Solution
Substitute 7×F for A and B, 6×F for C, 5×F for D, and 2×F for E in eq.1: 7×F + 7×F + 6×F + 5×F + 2×F + F = 28 which simplifies to 28×F = 28 Divide both sides of the above equation by 28: 28×F ÷ 28 = 28 ÷ 28 which means F = 1 making A = B = 7×F = 7 × 1 = 7 C = 6×F = 6 × 1 = 6 D = 5×F = 5 × 1 = 5 E = 2×F = 2 × 1 = 2 and ABCDEF = 776521