a-b-c-d-e-kids-are-in-ascending-order-of-heights-If-they-are-to-stand-in-a-circle-in-a-way-so-that-the-sum-of-difference-in-heights-of-adjacent-pairs-is-a-minimum-find-this-minimum-

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Question Number 155797 by ajfour last updated on 09/Oct/21

$$a,b,c,d,e\left(kids\right)areinascending$$$$orderofheights.Iftheyare$$$$tostandinacircleinawayso$$$$thatthesumof\mid differencein$$$$heightsofadjacentpairs\mid$$$$isaminimum,findthis$$$$minimum.$$

Commented by ajfour last updated on 09/Oct/21

$${(\mathrm{\Sigma }\mid \mathrm{a}-\mathrm{b}\mid )}_{\min }=?$$$$\mathrm{sir}?$$

Commented by mr W last updated on 05/Oct/21

$$a,b,c,d,earegiven,sothe$$$$sumisalways$$$$(e-d)+(d-c)+(c-b)+(b-a)=e-a$$

Commented by mr W last updated on 05/Oct/21

$$anycirclewithdiameter⩾eisok.$$

Commented by mr W last updated on 09/Oct/21

$$whatismeant?$$$$\mathrm{\Sigma }\mid a-b\mid =(b-a)+(c-b)+(d-c)+(e-d)?$$$$or$$$$\mathrm{\Sigma }\mid a-b\mid =(b-a)+(c-a)+(d-a)+(e-a)$$$$+(c-b)+(d-b)+(e-b)$$$$+(d-c)+(e-c)$$$$+(e-d)?$$

Commented by ajfour last updated on 09/Oct/21

$$\mid -3\mid =3$$$$\mid 3\mid =3$$$$\mid \mid \mathrm{is}\mathrm{the}\mathrm{modulus}\mathrm{function},\mathrm{sir}.$$$$\mathrm{say}\mathrm{a}=2,\mathrm{b}=4,\mathrm{c}=5,\mathrm{d}=7,\mathrm{e}=9$$$$\mathrm{Find}\mathrm{how}\mathrm{will}\mathrm{we}\mathrm{arrange}$$$$\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d},\mathrm{e}\mathrm{in}\mathrm{a}\mathrm{circle}\mathrm{so}\mathrm{that}$$$$\mathrm{y}=\mathrm{\Sigma }\mid {\mathrm{x}}_1-{\mathrm{x}}_2\mid \mathrm{\&}\mathrm{y}\mathrm{is}\mathrm{minimum}.$$$$\mathrm{say}\mathrm{if}\mathrm{we}\mathrm{arrange}\mathrm{circularly}$$$$\mathrm{like}\mathrm{eg}.\mathrm{acbeda}\mathrm{then}$$$${\mathrm{y}}_{\mathrm{eg}.}=\mid \mathrm{a}-\mathrm{c}\mid +\mid \mathrm{c}-\mathrm{b}\mid +\mid \mathrm{b}-\mathrm{e}\mid$$$$+\mid \mathrm{e}-\mathrm{d}\mid +\mid \mathrm{d}-\mathrm{a}\mid$$$${\mathrm{y}}_{\mathrm{eg}.}=\mid 2-5\mid +\mid 5-4\mid +\mid 4-9\mid$$$$+\mid 9-7\mid +\mid 7-2\mid$$$${\mathrm{y}}_{\mathrm{eg}.}=3+1+5+2+5=16$$$$\mathrm{but}\mathrm{we}\mathrm{need}{\mathrm{y}}_{\min }.$$$$\mathrm{what}\mathrm{if}\mathrm{we}\mathrm{arrange}\mathrm{in}$$$$\mathrm{ascending}\mathrm{order}\mathrm{itself}\mathrm{i}.\mathrm{e}.$$$$\mathrm{abcdea}$$$$\mathrm{then}{\mathrm{y}}_{\mathrm{ascend}}=2+1+2+2+7$$$$=14$$$$\mathrm{is}\mathrm{this}{\mathrm{y}}_{\min }?$$$$$$

Commented by mr W last updated on 09/Oct/21

$$nowiunderstandthequestion.i^{\prime }ll$$$$thinkaboutitagain.$$

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