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Produce-a-3-3-array-of-different-numbers-whose-all-the-rows-and-all-the-columns-make-AP-s-of-different-common-digfernces-

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Question Number 7423 by Rasheed Soomro last updated on 28/Aug/16
Produce a 3×3 array of different numbers whose all the rows and all the columns make AP′s of different common-digfernces.
$${Produce}\:{a}\:\mathrm{3}×\mathrm{3}\:\boldsymbol{{array}}\:{of}\:\boldsymbol{{different}}\:{numbers} \\ $$$${whose}\:{all}\:{the}\:\boldsymbol{{rows}}\:{and}\:{all}\:{the}\:\boldsymbol{{columns}}\:{make} \\ $$$$\boldsymbol{{AP}}'\boldsymbol{{s}}\:{of}\:\boldsymbol{{different}}\:\boldsymbol{{common}}-\boldsymbol{{digfernces}}. \\ $$
Commented by Yozzia last updated on 28/Aug/16
((1,2,3),(5,7,9),(9,(12),(15)) ) ((a,(a+d),(a+2d)),((a+b),((a+b)+c),((a+b)+2c)),((a+2b),((a+d)+2{(a+b)+c−(a+d)}),(a+2d+2{(a+b)+2c−a−2d})) ) ((a,(a+d),(a+2d)),((a+b),((a+b)+c),((a+b)+2c)),((a+2b),((a+2b)+(−d+2c)),((a+2b)+2(−d+2c))) ) The above matrix has all rows and columns each as APs. The common differences form the set { b , d , c , b+c−d , b+2c−2d , 2c−d }. # of distinct ways any two common−differences may be equal =5+4+3+2+1=15. If all members of the set are distinct, necessarily b≠c≠d. Also, b+c−d≠b+2c−2d , b+2c−2d≠2c−d and 2c−d≠b+c−d. ⇒ c−d≠0 , b−d≠0 and c≠b or c≠d, b≠d and c≠b. Likewise,(1) b≠b+c−d , b≠b+2c−2d, b≠2c−d or c≠d , c≠d , b≠2c−d; (2) d≠b+c−d , d≠b+2c−2d , d≠2c−d or 2d≠b+c, 3d≠b+2c, d≠c; (3) c≠b+c−d, c≠b+2c−2d , c≠2c−d or b≠d, 2d≠b+c, d≠c. I think it is sufficient then that b≠c≠d , b≠2c−d, 2d≠b+c and 3d≠b+2c to construct the matrix. Let a=0. ((0,d,(2d)),(b,(b+c),(b+2c)),((2b),(2b−d+2c),(2b−2d+4c)) ) Let d=2, b=1, c=3 ((0,2,4),(1,4,7),(2,6,(10)) ) Here 2d=b+c even though b≠c≠d and hence the matrix does not fit as an example. Let d=2,b=1, c=0. ∴ ((0,2,4),(1,1,1),(2,0,(−2)) ) This example works.
$$\begin{pmatrix}{\mathrm{1}}&{\mathrm{2}}&{\mathrm{3}}\\{\mathrm{5}}&{\mathrm{7}}&{\mathrm{9}}\\{\mathrm{9}}&{\mathrm{12}}&{\mathrm{15}}\end{pmatrix}\: \\ $$$$\begin{pmatrix}{{a}}&{{a}+{d}}&{{a}+\mathrm{2}{d}}\\{{a}+{b}}&{\left({a}+{b}\right)+{c}}&{\left({a}+{b}\right)+\mathrm{2}{c}}\\{{a}+\mathrm{2}{b}}&{\left({a}+{d}\right)+\mathrm{2}\left\{\left({a}+{b}\right)+{c}−\left({a}+{d}\right)\right\}}&{{a}+\mathrm{2}{d}+\mathrm{2}\left\{\left({a}+{b}\right)+\mathrm{2}{c}−{a}−\mathrm{2}{d}\right\}}\end{pmatrix}\: \\ $$$$\begin{pmatrix}{{a}}&{{a}+{d}}&{{a}+\mathrm{2}{d}}\\{{a}+{b}}&{\left({a}+{b}\right)+{c}}&{\left({a}+{b}\right)+\mathrm{2}{c}}\\{{a}+\mathrm{2}{b}}&{\left({a}+\mathrm{2}{b}\right)+\left(−{d}+\mathrm{2}{c}\right)}&{\left({a}+\mathrm{2}{b}\right)+\mathrm{2}\left(−{d}+\mathrm{2}{c}\right)}\end{pmatrix}\: \\ $$$${The}\:{above}\:{matrix}\:{has}\:{all}\:{rows}\:{and}\:{columns} \\ $$$${each}\:{as}\:{APs}.\: \\ $$$${The}\:{common}\:{differences}\:{form}\:{the}\:{set} \\ $$$$\left\{\:{b}\:,\:{d}\:,\:{c}\:,\:{b}+{c}−{d}\:,\:{b}+\mathrm{2}{c}−\mathrm{2}{d}\:,\:\mathrm{2}{c}−{d}\:\right\}. \\ $$$$#\:{of}\:{distinct}\:{ways}\:{any}\:{two}\:{common}−{differences}\: \\ $$$${may}\:{be}\:{equal}\:=\mathrm{5}+\mathrm{4}+\mathrm{3}+\mathrm{2}+\mathrm{1}=\mathrm{15}. \\ $$$${If}\:{all}\:{members}\:{of}\:{the}\:{set}\:{are}\:{distinct}, \\ $$$${necessarily}\:{b}\neq{c}\neq{d}. \\ $$$${Also},\:{b}+{c}−{d}\neq{b}+\mathrm{2}{c}−\mathrm{2}{d}\:,\:{b}+\mathrm{2}{c}−\mathrm{2}{d}\neq\mathrm{2}{c}−{d}\:{and}\:\mathrm{2}{c}−{d}\neq{b}+{c}−{d}. \\ $$$$\Rightarrow\:{c}−{d}\neq\mathrm{0}\:,\:{b}−{d}\neq\mathrm{0}\:{and}\:{c}\neq{b}\: \\ $$$${or}\:{c}\neq{d},\:{b}\neq{d}\:{and}\:{c}\neq{b}. \\ $$$${Likewise},\left(\mathrm{1}\right)\:{b}\neq{b}+{c}−{d}\:,\:{b}\neq{b}+\mathrm{2}{c}−\mathrm{2}{d},\:{b}\neq\mathrm{2}{c}−{d} \\ $$$${or}\:{c}\neq{d}\:,\:{c}\neq{d}\:,\:{b}\neq\mathrm{2}{c}−{d}; \\ $$$$\left(\mathrm{2}\right)\:{d}\neq{b}+{c}−{d}\:,\:{d}\neq{b}+\mathrm{2}{c}−\mathrm{2}{d}\:,\:{d}\neq\mathrm{2}{c}−{d} \\ $$$${or}\:\mathrm{2}{d}\neq{b}+{c},\:\mathrm{3}{d}\neq{b}+\mathrm{2}{c},\:{d}\neq{c}; \\ $$$$\left(\mathrm{3}\right)\:{c}\neq{b}+{c}−{d},\:{c}\neq{b}+\mathrm{2}{c}−\mathrm{2}{d}\:,\:{c}\neq\mathrm{2}{c}−{d} \\ $$$${or}\:{b}\neq{d},\:\mathrm{2}{d}\neq{b}+{c},\:{d}\neq{c}. \\ $$$${I}\:{think}\:{it}\:{is}\:{sufficient}\:{then}\:{that} \\ $$$${b}\neq{c}\neq{d}\:,\:{b}\neq\mathrm{2}{c}−{d},\:\mathrm{2}{d}\neq{b}+{c}\:{and}\:\mathrm{3}{d}\neq{b}+\mathrm{2}{c} \\ $$$${to}\:{construct}\:{the}\:{matrix}. \\ $$$${Let}\:{a}=\mathrm{0}. \\ $$$$\begin{pmatrix}{\mathrm{0}}&{{d}}&{\mathrm{2}{d}}\\{{b}}&{{b}+{c}}&{{b}+\mathrm{2}{c}}\\{\mathrm{2}{b}}&{\mathrm{2}{b}−{d}+\mathrm{2}{c}}&{\mathrm{2}{b}−\mathrm{2}{d}+\mathrm{4}{c}}\end{pmatrix}\: \\ $$$${Let}\:{d}=\mathrm{2},\:{b}=\mathrm{1},\:{c}=\mathrm{3} \\ $$$$\begin{pmatrix}{\mathrm{0}}&{\mathrm{2}}&{\mathrm{4}}\\{\mathrm{1}}&{\mathrm{4}}&{\mathrm{7}}\\{\mathrm{2}}&{\mathrm{6}}&{\mathrm{10}}\end{pmatrix}\:\:{Here}\:\mathrm{2}{d}={b}+{c}\:{even}\:{though}\:{b}\neq{c}\neq{d}\:{and}\:{hence} \\ $$$${the}\:{matrix}\:{does}\:{not}\:{fit}\:{as}\:{an}\:{example}. \\ $$$${Let}\:{d}=\mathrm{2},{b}=\mathrm{1},\:{c}=\mathrm{0}. \\ $$$$\therefore\:\begin{pmatrix}{\mathrm{0}}&{\mathrm{2}}&{\mathrm{4}}\\{\mathrm{1}}&{\mathrm{1}}&{\mathrm{1}}\\{\mathrm{2}}&{\mathrm{0}}&{−\mathrm{2}}\end{pmatrix}\:\:{This}\:{example}\:{works}. \\ $$$$ \\ $$
Commented by Rasheed Soomro last updated on 29/Aug/16
V N^ ice! G^( O^⌢ O_(⌣) ^(⌢) ) D analysis!
$${V}\:\mathcal{N}^{\:} {ice}!\:\:\:\mathcal{G}^{\:\:\underset{\smile} {\overset{\frown} {\mathcal{O}O}}} \mathcal{D}\:\:{analysis}! \\ $$

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