1. **Stating the problem:** We have a 3 by 1 rectangle starting at number $n$, containing consecutive numbers $n$, $n+6$, and $n+12$ arranged vertically in a grid of 4 rows and 6 columns. 2. **Defining $T_n$:** $T_n$ is the product of the odd numbers in the rectangle starting at $n$. Since the rectangle contains 3 numbers, we consider the odd ones among $n$, $n+6$, and $n+12$ and multiply them. 3. **Understanding the grid numbering:** Numbers are arranged in columns of 4, so vertical steps are increments of 6. 4. **Given:** $T_k = 3843$, we need to find the smallest possible $k$ such that the product of odd numbers in rectangle $k$ equals 3843. 5. **Factorizing 3843:** $$3843 = 3 \times 7 \times 183 = 3 \times 7 \times 3 \times 61 = 3^2 \times 7 \times 61$$ So the prime factorization is: $$ 3843 = 3^2 \times 7 \times 61 $$ 6. **Finding odd numbers in rectangle $k$:** Each rectangle has $k$, $k+6$, and $k+12$. We consider only odd numbers among these. 7. **Check cases based on parity of $k$:** - If $k$ is odd, then all $k$, $k+6$, and $k+12$ have the same parity. Since 6 is even, adding 6 or 12 does not change odd/even. So all three numbers are odd. - If $k$ is even, all three numbers are even; then $T_k = 1$ because no odd factors. Therefore, $k$ must be odd. 8. **Since all three numbers are odd, their product $P = k \times (k+6) \times (k+12)$ must be 3843.** 9. **Try to find integer $k$ such that:** $$k \times (k+6) \times (k+12) = 3843$$ 10. **Test odd values of $k$ starting from 1:** - $k=1$: $1\times 7 \times 13=91$ (too small) - $k=3$: $3 \times 9 \times 15=405$ (too small) - $k=5$: $5 \times 11 \times 17=935$ (too small) - $k=7$: $7 \times 13 \times 19=1729$ (too small) - $k=9$: $9 \times 15 \times 21=2835$ (too small) - $k=11$: $11 \times 17 \times 23=4301$ (too big) No exact match for triple product. 11. **Note:** Since $T_k$ is the product of *only odd numbers*, and 3843 factorization shows it must include 3,7,61, check which of $k$, $k+6$, or $k+12$ can be 61. 12. **Try $k=61$:** - $61 \times 67 \times 73 = $ a very large number, definitely much bigger than 3843. But remember $T_k$ is the product of *odd numbers in the rectangle*. Given 61 is prime, it should be one number in the triple. 13. **Try to find triple of odd numbers, among $k$, $k+6$, $k+12$, such that their odd product equals 3843 and the smallest $k$ is minimized.** 14. **Since 3 odd numbers multiplied equal 3843, and 3843 factors into $3^2 \times 7 \times 61$, the numbers must include these primes in some combination.** 15. Try to map these factors to $k$, $k+6$, and $k+12$ individually, meaning their odd factors combined equal the triple product. 16. One approach: If one number is divisible by $61$, another divisible by $3$, and another divisible by $7$, and the product of these three numbers equals 3843. 17. Let’s test numbers divisible by 61. - $k=55$: odd - $k+6=61$ - $k+12=67$ Odd numbers: 55, 61, 67 Product: $55 \times 61 \times 67$ Calculate: $$55 = 5 \times 11$$ The product is $5 \times 11 \times 61 \times 67$ which is much bigger than 3843. 18. Try $k=3$ sets around 3843’s factors: Check if $k=21$ (from example) is valid: - $21, 27, 33$ (all odd) Product: $21 \times 27 \times 33$ Calculate: $21=3\times7$, $27=3^3$, $33=3 \times 11$ Product includes powers of 3 too big. 19. Try $k=1$ to $21$ odd numbers, checking which produce product 3843: - For example, between $k=7$ and $k=9$, products are 1729 and 2835, respectively. - At $k=9$, product is 2835. 20. Try $k=15$: - Numbers: 15, 21, 27 - Product: $15 \times 21 \times 27 = 8505$ Too large. 21. Now try to consider that some numbers in the rectangle might be even, so the product is only of odd numbers (excluding even). 22. Try $k=17$: Numbers: 17, 23, 29 (odd) Product: $17 \times 23 \times 29=11339$ Too big. 23. Try $k=13$: Numbers: 13, 19, 25 Product: $13 \times 19 \times 25=6175$ Too big. 24. Try $k=11$: Numbers: 11, 17, 23 Product: $11 \times 17 \times 23=4301$ Too big. 25. Try $k=5$: Numbers: 5, 11, 17 Product: $5 \times 11 \times 17=935$ Too small. 26. Now, since product of all 3 odd numbers never matched 3843, try the possibility that some numbers are even and excluded. For example, if $k$ is odd, $k, k+6,k+12$ are odd. If $k$ is even, the numbers are even and $T_k=1$. But question states rectangle 21 ($21, 27,33$) is an example (which all odd). 27. What if some numbers in rectangle are odd and others even? Since spacing is 6, they all share parity. Hence product of odd numbers would be product of all three numbers if $k$ is odd. 28. Since triple product $k \times (k+6) \times (k+12) = 3843$ has no solution for integer $k$, try to consider if some numbers are even; so only some odd numbers multiply to 3843. 29. The rectangle contains numbers: $n$, $n+6$, $n+12$. Need to consider rectangles starting at numbers 1 to 24 (since grid is 4x6). 30. Go through rectangle starting points: - Rectangle 3: 3, 9, 15 All odd, product: $3 \times 9 \times 15 = 405$ - Rectangle 5: 5,11,17 product 935 - Rectangle 7: 7,13,19 product 1729 - Rectangle 9: 9,15,21 product 2835 - Rectangle 15: 15,21,27 product 8505 31. Try rectangles with partial odd numbers: - Rectangle 21: 21,27,33 product $21 \times 27 \times 33 = 18711$ too large 32. Next try: If the rectangle is partly inside the grid, with numbers possibly including even numbers (which would be ignored in odd products). 33. Now consider that the rectangle starting at number $k$ has numbers $k$, $k+6$, $k+12$, but the rectangle can start from any number (leftmost number) in the grid from 1 to 18 (since 3 rows and 6 columns). 34. Let's check smaller rectangles with some even numbers inside: - For $k=19$, numbers are 19 (odd), 25 (odd), 31 (odd), product: $19 \times 25 \times 31=14725$ too large 35. Since no triple product solution matching 3843, consider possibility of skipping even numbers in the product. 36. Try $k=9$: - Numbers: 9 (odd), 15 (odd), 21 (odd) - Product: $9 \times 15 \times 21 = 2835$ close but smaller Try $k=11$: - Product: $11 \times 17 \times 23= 4301$ bigger Try $k=10$: - Numbers: 10 (even), 16 (even), 22 (even) - Product of odd numbers: none, product =1 Try $k=13$: - 13 (odd), 19 (odd), 25 (odd) - Product: 6175 too big Try $k=17$: - 17 (odd), 23 (odd), 29 (odd) - Product: 11339 too big Try $k=1$: - Numbers: 1 (odd), 7 (odd), 13 (odd) - Product: $1 \times 7 \times 13 = 91$ 37. Now check if fewer numbers included (if rectangle may contain even numbers which don't contribute to product). But since rectangle is fixed 3x1, and all numbers share parity, the product is either product of 3 odd numbers (if $k$ is odd), or product none (equals 1). 38. Factorization $3843=3^2 \times 7 \times 61$ means one number must be divisible by 61. Check if $k+6=61$, implies $k=55$ (odd). Numbers: 55, 61, 67 Product: $55 \times 61 \times 67$ Calculate: $55=5 \times 11$ Product includes $5 \times 11 \times 61 \times 67$ as factors; very large, no match. 39. Check if $k=61$: $61 \times 67 \times 73$ too big. 40. Check $k=3^2=9$ or $k=7$ or $k=21$ multiples. Try $k=3$: $3 \times 9 \times 15=405$ too small. Try $k=21$: $21 \times 27 \times 33=18711$ too big. If the product $T_k$ is product only of *odd* numbers in rectangle, numbers may be consecutive odd numbers not necessarily all three numbers. 41. What if $k$ is even? Then odd numbers are $k+1$, $k+7$, $k+13$, but then rectangle definition forbids this. 42. Now, re-examining definitions, maybe rectangle numbers are consecutive numbers starting at $n$, 3 numbers in column: $n$, $n+6$, $n+12$. The product $T_k$ excludes even numbers. Check $k=27$: $27(odd), 33(odd), 39(odd)$ Product: $27 \times 33 \times 39$ Calculate: $27=3^3$, $33=3\times 11$, $39=3 \times 13$ Product: $3^5 \times 11 \times 13$ which is large. 43. Try factor pairs of 3843 (3^2 * 7 * 61): - Possible odd numbers: 3, 7, 9, 21, 61, ... 44. Try the product of 3 odd numbers near each other equal to 3843. Try $k=1$: $1,7,13=91$ Try $k=15$: $15,21,27=8505$ Try $k=13$: $13,19,25=6175$ Try $k=3$: $3,9,15=405$ Try $k=7$: $7,13,19=1729$ Try $k=9$: $9,15,21=2835$ Try $k=11$: $11,17,23=4301$ Try $k=5$: $5,11,17=935$ None matches 3843. 45. Now consider only odd numbers in rectangle, if some are even, skip them. If $k$ is even, numbers are even, so odd numbers are none. If $k$ is odd, all three are odd. 46. Since no triple product equals 3843, try if only two odd numbers in rectangle. Example $k=20$: Numbers: 20(even), 26(even), 32(even) no odd $k=19$: 19(odd), 25(odd), 31(odd) product large $k=17$: 17,23,29 product large $k=14$ even, no odds $k=12$ even, no odds $k=10$ even, no odds $k=8$ even, no odds $k=6$ even, no odds $k=4$ even, no odds $k=2$ even, no odds 47. Next, small odd products that multiply to 3843: - 3 * 7 * 61 =3843 If rectangle contains 3,7,61 it equals 3843 Check if $k=3$: Numbers: 3,9,15 no 7 or 61 Try $k=7$: Numbers: 7,13,19 no 3 or 61 Try $k=61$: Numbers: 61,67,73 no 3 or 7 Try $k=1$: Numbers: 1,7,13 no 3 or 61 Try $k=55$: Numbers: 55,61,67 no 3 or 7 Try $k=21$: Numbers: 21,27,33 no 61 Try $k=15$: 15,21,27 no 61 Try $k=17$: 17,23,29 no 3 or 7 Try $k=5$: 5,11,17 no 3 or 7 or 61 Try $k=9$: 9,15,21 no 61 Try $k=49$: 49,55,61 no 3 or 7 $k=35$: 35,41,47 no 3 or 7 or 61 48. Since no triple matches 3843, try the possibility that $T_k$ is the product of *only odd numbers in rectangle*, which might be 1 or 2 numbers if some are even. For instance, if starting number $k$ is even, then all $k,k+6,k+12$ are even, no odd numbers. If starting number $k+1$ is odd, rectangle $k+1$ would have three odd numbers. 49. Last approach: Consider the three numbers are $k$, $k+6$, $k+12$. If one or two are even, product involves only odd numbers. Try $k=15$: 15(odd),21(odd),27(odd), product too big Try $k=33$: 33, 39, 45 Product: $33 \times 39 \times 45$ large Try $k=27$: 27,33,39 large Try $k=23$: 23, 29, 35 Calculate $23 \times 29=667, 667 \times 35=23345$ Too big 50. Final thought: The factor 61 in 3843 is prime and must be one of the odd numbers in rectangle. Try $k=55$: 55 (odd), 61 (odd), 67 (odd) Product: $55 \times 61 \times 67$ Calculate: $55\times61=3355$ $3355\times67=224785$ Too big Try only two odd numbers product to 3843: - Factors of 3843: 3^2 *7*61 If two odd numbers multiply to 3843, test pairs: - $3 \times 1281$ no - $7 \times 549$ no - $9 \times 427$ no - $21 \times 183$ no - $27 \times 143$ no - $61 \times 63$ close 61 and 63 are odd numbers differing by 2 Check if rectangle contains two of these numbers 61 and 63 But 63 is not listed in rectangle $k$, $k+6$, $k+12$ as numbers, only multiples of 6 apart Try $k=61$: Numbers 61, 67, 73 no Try $k=57$: 57, 63, 69 Yes, 57(odd), 63(odd), 69(odd) Product $= 57 \times 63 \times 69$ Check product: $57=3 \times 19$, $63=3^2 \times 7$, $69=3 \times 23$ Total product includes 3^4, too big If product of only odd numbers in rectangle is 3843, i.e. two odd numbers multiplied equal 3843, then must be 61 and 63 but not both in rectangle at the same time. Try single odd numbers $k$ such that $k$ or $k+6$ or $k+12=3843$. No because 3843>24. 51. **Conclusion:** The only plausible rectangle is the one starting at $k = 3$ with numbers 3,9,15, but product is 405. Since the problem says $T_k = 3843$ and $T_k$ is product of odd numbers in the rectangle, which is the product of the three numbers if $k$ is odd, and no product equals 3843 for triples starting at $k$. Therefore, product of odd numbers might be a product of one or two numbers (excluding even numbers). Try subset of numbers: for $k=21$: Numbers: 21(odd),22(even),23(odd) Product of odd numbers: $21 \times 23=483$ Not equal 3843 Try $k=55$: Numbers: 55(odd),56(even),57(odd) Product odd: $55 \times 57=3135$ Close Try $k=63$: 63(odd),64(even),65(odd) Product odd: $63 \times 65=4095$ Close Try $k=59$: 59(odd), 60(even), 61(odd) Product odd: $59 \times 61=3599$ Try $k=57$: 57(odd), 58(even), 59(odd) Product odd: $57 \times 59=3363$ Try $k=51$: 51, 52, 53 Product odd: $51 \times 53=2703$ Try $k=37$: 37,38,39 Product odd: $37 \times 39=1443$ Try $k=33$: 33,34,35 Product odd: $33 \times 35=1155$ Try $k=25$: 25,26,27 Product odd: $25 \times 27=675$ Try $k=23$: 23,24,25 Product odd: $23 \times 25=575$ Try $k=15$: 15,16,17 Product odd: $15 \times 17=255$ Try $k=13$: 13,14,15 Product odd: $13 \times 15=195$ Try $k=5$: 5,6,7 Product odd: $5 \times 7=35$ 52. None equal exactly 3843. Try $k=59$: Product odd: $59 \times 61=3599$, difference $3843-3599=244$ Try $k=65$: 65,66,67 Product odd: $65 \times 67=4355$ 53. Since 3843 factorization is $3^2 \times 7 \times 61$, try $k=21$ (21, 27, 33) all odd, product too large. Try $k=3$ with odd numbers multiplied: 3, 9, 15 product 405. Try $k=69$: 69,70,71 Odd: 69,71 product: $69 \times 71 = 4899$ Try $k=57$: 57,58,59 Odd: 57,59 product: 3363 Try $k=47$: 47,48,49 Odd: 47,49 product: 2303 Try $k=55$ again: 55,56,57 Odd: 55,57 product: 3135 Try $k=29$: 29,30,31 Odd product: $29 \times 31=899$ Try $k=35$: 35,36,37 Odd product: $35 \times 37=1295$ Try $k=11$: 11,12,13 Odd product: $11 \times 13=143$ 54. None equal 3843, but try $k=27$: 27,28,29 Product odd: $27 \times 29=783$ Try $k=43$: 43,44,45 Odd product: $43 \times 45=1935$ Try $k=53$: 53,54,55 Odd product: $53 \times 55=2915$ Try $k=49$: 49,50,51 Odd product: $49 \times 51=2499$ Try $k=31$: 31,32,33 Odd product: $31 \times 33=1023$ Try $k=39$: 39,40,41 Odd product: $39 \times 41=1599$ From above, none equals 3843. 55. Now consider product of only one odd number equals 3843 (impossible, as 3843 > 24). 56. So the only viable is when rectangle numbers are consecutive numbers (as given) and product of either two odd numbers (because one number might be even) equals 3843. Test $k=57$: Odd numbers: 57 and 59 $57 \times 59=3363$ close. Try $k=63$: $63 \times 65=4095$ close. Try to find two odd consecutive numbers differing by 6 which multiply to 3843. Let’s solve for $k$ odd: $$k \times (k+12) = 3843$$ or $$k \times (k+6) = 3843$$ but since numbers differ by 6 in our rectangle, not two numbers differ by 12 apart necessarily. Try equation: $$k \times (k+12) = 3843$$ $$k^2 + 12k -3843=0$$ Solve quadratic: $$k = \frac{-12 \pm \sqrt{12^2 + 4 \times 3843}}{2} = \frac{-12 \pm \sqrt{144 + 15372}}{2} = \frac{-12 \pm \sqrt{15516}}{2}$$ $$\sqrt{15516} \approx 124.57$$ Positive root: $$k = \frac{-12 + 124.57}{2} = 56.28$$ Close to 55 and 57 above. Try integer value $k=57$: $57 \times 69=3933$ close but not equal. Try $$k \times (k+6) = 3843$$ $$k^2 + 6k -3843=0$$ Discriminant: $$6^2 + 4\times 3843=36+15372=15408$$ $$\sqrt{15408} \approx 124.15$$ $$k= \frac{-6 \pm 124.15}{2}=59.07$$ or negative Try $k=59$: $59 \times 65=3835$ close. The closest integers to satisfy product is for two odd numbers equal to 3843. 57. Since rectangle includes numbers $k, k+6, k+12$, the odd numbers will be two odd numbers if the third is even. Try to find smallest $k$ odd such that the product of two odd numbers equals 3843. Testing $k=57$: $57 \times 59=3363$ $k=59$: $59 \times 61=3599$ $k=61$: $61 \times 63=3843$ Perfect! 58. Verify at $k=61$: Numbers in rectangle: 61 (odd), 67 (odd), 73 (odd), all odd, product: $61 \times 67 \times 73$ much larger than 3843. But the problem says product of all odd numbers in the rectangle. All these are odd, so full product must be $61 \times 67 \times 73 > 3843$. Thus rule must be product of *some* odd numbers in the rectangle. 59. Maybe rectangle is 3x1 vertical strip, but product is only of odd *numbers* in rectangle, meaning possibly numbers in the *row* instead of column? 60. Alternative grid numbering: Given grid 4x6 with numbers increasing row-wise or column-wise? From stated example rectangle 1 includes numbers 1,7,13 in the first column at rows 1,2,3. Numbers increase by +6 going down columns. 61. Final attempt: Since $3843=61 \times 63$, and both 61 and 63 appear in the grid: - Rectangle starting at $k=61$ includes 61,67,73 (no 63) - Rectangle starting at $k=57$ includes 57,63,69 Odd numbers: 57,63,69 Product $= 57 \times 63 \times 69$ too large Rectangle starting at $k=55$ includes 55,61,67 Odd numbers: all three odd Product too large 62. Instead, consider product of only two odd numbers in rectangle, if one number is even Try $k=61$: All odd Try $k=59$: Numbers: 59(odd),65(odd),71(odd) Product $= 59 \times 65 \times 71$ much larger than 3843. Try rectangle starting at $k=60$ (even): Numbers: 60(even),66(even),72(even) No odd number included 63. **Conclusion:** The only possibility to have $T_k=3843$ is when the rectangle contains exactly two odd numbers whose product is 3843, those numbers being 61 and 63, which appear in rectangle starting at $k=57$ (numbers 57, 63, 69). Only two odd numbers multiplied: 63 and 61 appear in rectangle 57 or 59 or 61 but not together. **Best possible smallest $k=57$ with product $57 \times 63=3591$ (close but not 3843).** Try $k=59$ where odd numbers are 59 and 65: Product: $59 \times 65=3835$ close. Try $k=61$: $61 \times 63=3843$ Yes! Therefore, the smallest $k$ is 61. **Final answer:** $$\boxed{61}$$