Puzzle for May 7, 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.
Scratchpad
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Hint #1
In eq.5, replace D with C + F (from eq.2): A + C + F = B + C + F – F which becomes A + C + F = B + C Subtract C from both sides of the equation above: A + C + F – C = B + C – C which becomes eq.5a) A + F = B
Hint #2
In eq.6, replace B with A + F (from eq.5a), and D with C + F (from eq.2): A + F + C + F – A = A + E – F which becomes C + 2×F = A + E – F Add F to both sides of the above equation: C + 2×F + F = A + E – F + F which becomes eq.6a) C + 3×F = A + E
Hint #3
eq.3 may be written as: B + F = A + E – B + C In the above equation, substitute C + 3×F for A + E (from eq.6a): B + F = C + 3×F – B + C which becomes B + F = 2×C + 3×F – B Subtract F from both sides, and add B to both sides: B + F – F + B = 2×C + 3×F – B – F + B which becomes 2×B = 2×C + 2×F Divide both sides by 2: 2×B ÷ 2 = (2×C + 2×F) ÷ 2 which becomes eq.3a) B = C + F
Hint #4
In eq.5a, substitute C + F for B (from eq.3a): A + F = C + F Subtract F from each side of the equation above: A + F – F = C + F – F which makes A = C
Hint #5
Substitute A for C in eq.6a: A + 3×F = A + E Subtract A from both sides of the equation above: A + 3×F – A = A + E – A which makes 3×F = E
Hint #6
In eq.2, substitute B for C + F (from eq.3a): B = D
Hint #7
Substitute B for D, 3×F for E, and A for C in eq.4: B + 3×F = A + A – 3×F – F which becomes B + 3×F = 2×A – 4×F Add 4×F to both sides of the above equation: B + 3×F + 4×F = 2×A – 4×F + 4×F which becomes eq.4a) B + 7×F = 2×A
Hint #8
Substitute A + F for B (from eq.5a) in eq.4a: A + F + 7×F = 2×A which becomes A + 8×F = 2×A Subtract A from each side of the equation above: A + 8×F – A = 2×A – A which makes 8×F = A and also makes C = A = 8×F
Hint #9
Substitute 8×F for A in eq.5a: 8×F + F = B which makes 9×F = B and also makes D = B = 9×F
Solution
Substitute 8×F for A and C, 9×F for B and D, and 3×F for E in eq.1: 8×F + 9×F + 8×F + 9×F + 3×F + F = 38 which simplifies to 38×F = 38 Divide both sides of the above equation by 38: 38×F ÷ 38 = 38 ÷ 38 which means F = 1 making A = C = 8×F = 8 × 1 = 8 B = D = 9×F = 9 × 1 = 9 E = 3×F = 3 × 1 = 3 and ABCDEF = 898931