Puzzle for January 4, 2023 ( )
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.6, replace A + B with C + D + F (from eq.5): C + D - F = C + D + F - D which becomes C + D - F = C + F In the above equation, subtract C from both sides, and add F to both sides: C + D - F - C + F = C + F - C + F which becomes D = 2×F
Hint #2
In eq.2, replace D with 2×F: F - C = C - 2×F Add C and 2×F to both sides of the equation above: F - C + C + 2×F = C - 2×F + C + 2×F which makes 3×F = 2×C Divide both sides by 2: 3×F ÷ 2 = 2×C ÷ 2 which makes 1½×F = C
Hint #3
In eq.3, substitute 1½×F for C: 1½×F - E = B - 1½×F Add 1½×F to both sides of the above equation: 1½×F - E + 1½×F = B - 1½×F + 1½×F which becomes eq.3a) 3×F - E = B
Hint #4
Substitute 3×F - E for B (from eq.3a), 1½×F for C, and 2×F for D in eq.4: 3×F - E + 1½×F = 2×F + E + F which becomes 4½×F - E = 3×F + E In the above equation, add E to both sides, and subtract 3×F from both sides: 4½×F - E + E - 3×F = 3×F + E + E - 3×F which becomes 1½×F = 2×E Divide both sides by 2: 1½×F ÷ 2 = 2×E ÷ 2 which makes ¾×F = E
Hint #5
Substitute ¾×F for E in eq.3a: 3×F - ¾×F = B which makes 2¼×F = B
Hint #6
Substitute 2¼×F for B, 1½×F for C, and 2×F for D in eq.5: A + 2¼×F = 1½×F + 2×F + F which becomes A + 2¼×F = 4½×F Subtract 2¼×F from each side of the above equation: A + 2¼×F - 2¼×F = 4½×F - 2¼×F which makes A = 2¼×F
Solution
Substitute 2¼×F for A and B, 1½×F for C, 2×F for D, and ¾×F for E in eq.1: 2¼×F + 2¼×F + 1½×F + 2×F + ¾×F + F = 39 which simplifies to 9¾×F = 39 Divide both sides of the above equation by 9¾: 9¾×F ÷ 9¾ = 39 ÷ 9¾ which means F = 4 making A = B = 2¼×F = 2¼ × 4 = 9 C = 1½×F = 1½ × 4 = 6 D = 2×F = 2 × 4 = 8 E = ¾×F = ¾ × 4 = 3 and ABCDEF = 996834