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Question Number 19704 by Tinkutara last updated on 14/Aug/17

$$\mathrm{What}\mathrm{is}\mathrm{the}\mathrm{sum}\left(\mathrm{in}\mathrm{base}10\right)\mathrm{of}\mathrm{all}\mathrm{the}$$$$\mathrm{natural}\mathrm{numbers}\mathrm{less}\mathrm{than}64\mathrm{which}$$$$\mathrm{have}\mathrm{exactly}\mathrm{three}\mathrm{ones}\mathrm{in}\mathrm{their}\mathrm{base}2$$$$\mathrm{representation}?$$

Answered by mrW1 last updated on 15/Aug/17

$$‘‘1^{″}\mathrm{means}1,2,4,8,16,32\mathrm{from}$$$$\mathrm{right}\mathrm{to}\mathrm{left}.$$$$\mathrm{with}\mathrm{three}‘‘1^{″}\mathrm{we}\mathrm{can}\mathrm{get}$$$$\mathrm{following}\mathrm{possible}\mathrm{numbers}:$$$$32+16+8$$$$32+16+4$$$$32+16+2$$$$32+16+1\Rightarrow \mathrm{\Sigma }=4\times 32+5\times 16-1=207$$$$32+8+4$$$$32+8+2$$$$32+8+1\Rightarrow \mathrm{\Sigma }=3\times 32+4\times 8-1=127$$$$32+4+2$$$$32+4+1\Rightarrow \mathrm{\Sigma }=2\times 32+3\times 4-1=75$$$$32+2+1\Rightarrow 35$$$$16+8+4$$$$16+8+2$$$$16+8+1\Rightarrow \mathrm{\Sigma }=3\times 16+4\times 8-1=79$$$$16+4+2$$$$16+4+1\Rightarrow \mathrm{\Sigma }=2\times 16+3\times 4-1=43$$$$16+2+1\Rightarrow 19$$$$8+4+2$$$$8+4+1\Rightarrow \mathrm{\Sigma }=2\times 8+3\times 4-1=27$$$$8+2+1\Rightarrow 11$$$$4+2+1\Rightarrow 7$$$$$$$$\mathrm{\Sigma }=207+127+75+35+79+43+19+27+11+7$$$$=630$$

Commented by Tinkutara last updated on 15/Aug/17

$$\mathrm{Thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{Sir}!$$

Commented by mrW1 last updated on 16/Aug/17

$$\mathrm{there}\mathrm{is}\mathrm{an}\mathrm{easier}\mathrm{way}\mathrm{to}\mathrm{find}\mathrm{the}\mathrm{sum}$$$$\mathrm{without}\mathrm{knowing}\mathrm{each}\mathrm{number}:$$$$$$$$2\mathrm{\Sigma }=(1+2+4+8+16+32)\times 5\times 4=1260$$$$\Rightarrow \mathrm{\Sigma }=1260/2=630$$

Commented by ajfour last updated on 16/Aug/17

$$111$$$$1110,\mathrm{and}2\mathrm{other}4\mathrm{digit}\mathrm{ones}$$$$11100,\mathrm{and}5\mathrm{other}5\mathrm{digit}\mathrm{ones}$$$$111000,\mathrm{and}9\mathrm{other}6\mathrm{digit}\mathrm{ones}$$$$111+\mathrm{\Sigma }1110+\mathrm{\Sigma }11100+\mathrm{\Sigma }111000$$$$=7+3\times 8+2(4+2+1)+$$$$6\times 16+3(8+4+2+1)+$$$$10\times 32+4(16+8+4+2+1)$$$$=7+24+14+96+45+320+124$$$$=630.$$

Commented by Tinkutara last updated on 16/Aug/17

$$\mathrm{Thank}\mathrm{you}\mathrm{both}\mathrm{mrW}1\mathrm{and}\mathrm{ajfour}.$$

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