Puzzle for June 1, 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.3, replace A with C + D + F (from eq.6): C + D + E = C + D + F + F Subtract C and D from each side of the above equation: C + D + E – C – D = C + D + F + F – C – D which simplifies to E = 2×F
Hint #2
In eq.5, replace E with 2×F: A – 2×F = B + F Add 2×F to both sides of the above equation: A – 2×F + 2×F = B + F + 2×F which becomes eq.5a) A = B + 3×F
Hint #3
In eq.2, substitute B + 3×F for A (from eq.5a): B + C = B + 3×F Subtract B from both sides of the equation above: B + C – B = B + 3×F – B which makes C = 3×F
Hint #4
Substitute 3×F for C in eq.4: D = 3×F + F which makes D = 4×F
Hint #5
Substitute 3×F for C, 4×F for D, and 2×F for E in eq.3: 3×F + 4×F + 2×F = A + F which becomes 9×F = A + F Subtract F from each side of the equation above: 9×F – F = A + F – F which makes 8×F = A
Hint #6
Substitute 3×F for C, and 8×F for A in eq.2: B + 3×F = 8×F Subtract 3×F from each side of the above equation: B + 3×F – 3×F = 8×F – 3×F which makes B = 5×F
Solution
Substitute 8×F for A, 5×F for B, 3×F for C, 4×F for D, and 2×F for E in eq.1: 8×F + 5×F + 3×F + 4×F + 2×F + F = 23 which simplifies to 23×F = 23 Divide both sides of the above equation by 23: 23×F ÷ 23 = 23 ÷ 23 which means F = 1 making A = 8×F = 8 × 1 = 8 B = 5×F = 5 × 1 = 5 C = 3×F = 3 × 1 = 3 D = 4×F = 4 × 1 = 4 E = 2×F = 2 × 1 = 2 and ABCDEF = 853421