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Question Number 113355 by Aina Samuel Temidayo last updated on 12/Sep/20
$$\mathrm{A}\:\mathrm{rectangular}\:\mathrm{cardboard}\:\mathrm{is}\:\mathrm{8cm}\:\mathrm{long} \\ $$$$\mathrm{and}\:\mathrm{6cm}\:\mathrm{wide}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{least} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{beads}\:\mathrm{you}\:\mathrm{can}\:\mathrm{arrange}\:\mathrm{on} \\ $$$$\mathrm{the}\:\mathrm{board}\:\mathrm{such}\:\mathrm{that}\:\mathrm{there}\:\mathrm{are}\:\mathrm{at}\:\mathrm{least} \\ $$$$\mathrm{two}\:\mathrm{of}\:\mathrm{the}\:\mathrm{beads}\:\mathrm{that}\:\mathrm{are}\:\mathrm{less}\:\mathrm{than} \\ $$$$\sqrt{\mathrm{10}}\mathrm{cm}\:\mathrm{apart}. \\ $$
Commented by Aina Samuel Temidayo last updated on 12/Sep/20
$$\mathrm{Solution}\:\mathrm{please}? \\ $$
Answered by mr W last updated on 13/Sep/20
Commented by mr W last updated on 13/Sep/20
$${the}\:{points}\:{on}\:{the}\:{grid}\:{are}\:\sqrt{\mathrm{10}}\:{apart} \\ $$$${from}\:{each}\:{other}. \\ $$$${with}\:{a}\:{rectangle}\:{of}\:\mathrm{6}×\mathrm{8}\:{we}\:{can}\:{cover} \\ $$$${at}\:{most}\:\mathrm{9}\:{points}. \\ $$$${it}\:{means}\:{if}\:{we}\:{put}\:\mathrm{10}\:{points}\:{or}\:{more} \\ $$$${on}\:{the}\:\mathrm{6}×\mathrm{8}\:{board}\:{there}\:{are}\:{at}\:{least}\:{two} \\ $$$${points}\:{which}\:{are}\:{less}\:{than}\:\sqrt{\mathrm{10}}\:{apart} \\ $$$${from}\:{each}\:{other}. \\ $$