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Question Number 202037 by syamilkamil1 last updated on 19/Dec/23
  A board has 2, 4, and 6 written on it. One repeatedly chooses values ​​(not necessarily different) for x, y, and z from the board, and writes xyz + xy + yz + zx + x + y + z if and only if those numbers are not already on the board and are also less than or equals 2013. The person repeats this process until no more numbers can be written. How many numbers will be written at the end of this process?
$$ \\ $$A board has 2, 4, and 6 written on it. One repeatedly chooses values ​​(not necessarily different) for x, y, and z from the board, and writes xyz + xy + yz + zx + x + y + z if and only if those numbers are not already on the board and are also less than or equals 2013. The person repeats this process until no more numbers can be written. How many numbers will be written at the end of this process?
Answered by nikif99 last updated on 20/Dec/23
Just 20. See image for details.  Proposition: If a number n>=223 is   written, it ceases procedure with this  n because next result exceeds 2013.  (x,y,z)=(2,2,223)→2×2×223+2×2+  2×223+2×223+2+2+223=2015.  In the region A1..C11 are 10 possible  cases when only 2,4,6 are written (see  col. E for start).  For each of these cases, application of  the given formula for each row gives  as result column N, so the respective  number can be written on the board.  ★For the 2nd round you apply each  number from the range N2..N11 to the  cell N1 (in the example, N6=62), which  copies possible triplettes to the range  A13..C22.  So, you have 10 new cases to apply the  formula (if you omit 62 and use 2,4,6  only, you are driven to a repetition).  New triplettes of rows 13 to 22 lead to  results N13..N22. Not all results are  acceptable since some of them (marked  in red) are >2013.  Accepted results are written vertically  in cols F to M (in our example, H).  Repeating the same procedure from ★  for all numbers in N2..N11 you get  results in cols F to M.  It is impossible to follow 3rd round  because all numbers in cols F to M  are >223 (see proposition above).  Finally, collecting all produced numbers  you get final result in A24..H28.
$${Just}\:\mathrm{20}.\:{See}\:{image}\:{for}\:{details}. \\ $$$${Proposition}:\:{If}\:{a}\:{number}\:{n}>=\mathrm{223}\:{is}\: \\ $$$${written},\:{it}\:{ceases}\:{procedure}\:{with}\:{this} \\ $$$${n}\:{because}\:{next}\:{result}\:{exceeds}\:\mathrm{2013}. \\ $$$$\left({x},{y},{z}\right)=\left(\mathrm{2},\mathrm{2},\mathrm{223}\right)\rightarrow\mathrm{2}×\mathrm{2}×\mathrm{223}+\mathrm{2}×\mathrm{2}+ \\ $$$$\mathrm{2}×\mathrm{223}+\mathrm{2}×\mathrm{223}+\mathrm{2}+\mathrm{2}+\mathrm{223}=\mathrm{2015}. \\ $$$${In}\:{the}\:{region}\:{A}\mathrm{1}..{C}\mathrm{11}\:{are}\:\mathrm{10}\:{possible} \\ $$$${cases}\:{when}\:{only}\:\mathrm{2},\mathrm{4},\mathrm{6}\:{are}\:{written}\:\left({see}\right. \\ $$$$\left.{col}.\:{E}\:{for}\:{start}\right). \\ $$$${For}\:{each}\:{of}\:{these}\:{cases},\:{application}\:{of} \\ $$$${the}\:{given}\:{formula}\:{for}\:{each}\:{row}\:{gives} \\ $$$${as}\:{result}\:{column}\:{N},\:{so}\:{the}\:{respective} \\ $$$${number}\:{can}\:{be}\:{written}\:{on}\:{the}\:{board}. \\ $$$$\bigstar{For}\:{the}\:\mathrm{2}{nd}\:{round}\:{you}\:{apply}\:{each} \\ $$$${number}\:{from}\:{the}\:{range}\:{N}\mathrm{2}..{N}\mathrm{11}\:{to}\:{the} \\ $$$${cell}\:{N}\mathrm{1}\:\left({in}\:{the}\:{example},\:{N}\mathrm{6}=\mathrm{62}\right),\:{which} \\ $$$${copies}\:{possible}\:{triplettes}\:{to}\:{the}\:{range} \\ $$$${A}\mathrm{13}..{C}\mathrm{22}. \\ $$$${So},\:{you}\:{have}\:\mathrm{10}\:{new}\:{cases}\:{to}\:{apply}\:{the} \\ $$$${formula}\:\left({if}\:{you}\:{omit}\:\mathrm{62}\:{and}\:{use}\:\mathrm{2},\mathrm{4},\mathrm{6}\right. \\ $$$$\left.{only},\:{you}\:{are}\:{driven}\:{to}\:{a}\:{repetition}\right). \\ $$$${New}\:{triplettes}\:{of}\:{rows}\:\mathrm{13}\:{to}\:\mathrm{22}\:{lead}\:{to} \\ $$$${results}\:{N}\mathrm{13}..{N}\mathrm{22}.\:{Not}\:{all}\:{results}\:{are} \\ $$$${acceptable}\:{since}\:{some}\:{of}\:{them}\:\left({marked}\right. \\ $$$$\left.{in}\:{red}\right)\:{are}\:>\mathrm{2013}. \\ $$$${Accepted}\:{results}\:{are}\:{written}\:{vertically} \\ $$$${in}\:{cols}\:{F}\:{to}\:{M}\:\left({in}\:{our}\:{example},\:{H}\right). \\ $$$${Repeating}\:{the}\:{same}\:{procedure}\:{from}\:\bigstar \\ $$$${for}\:{all}\:{numbers}\:{in}\:{N}\mathrm{2}..{N}\mathrm{11}\:{you}\:{get} \\ $$$${results}\:{in}\:{cols}\:{F}\:{to}\:{M}. \\ $$$${It}\:{is}\:{impossible}\:{to}\:{follow}\:\mathrm{3}{rd}\:{round} \\ $$$${because}\:{all}\:{numbers}\:{in}\:{cols}\:{F}\:{to}\:{M} \\ $$$${are}\:>\mathrm{223}\:\left({see}\:{proposition}\:{above}\right). \\ $$$${Finally},\:{collecting}\:{all}\:{produced}\:{numbers} \\ $$$${you}\:{get}\:{final}\:{result}\:{in}\:{A}\mathrm{24}..{H}\mathrm{28}. \\ $$
Commented by nikif99 last updated on 20/Dec/23

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