14.1000 Multiple sticks can be arranged into Figure 1 (a rectangle in one row), Figure 2 (a rectangle in two rows), and Figure 3 (a square), with no sticks left over in any case. What is the minimum number of sticks $\qquad$ needed?



Translating the text into English while preserving the original text's line breaks and format, the result is as above.
【Answer】6, 5, 4, 3, 2
【Analysis】Key Point: Sequence Pattern, Remainder
Let the first figure be $1 \times k$ small squares, the second figure be $2 \times m$ small squares, and the third figure be $n \times n$ small squares. Therefore, we have $3 k+1=5 m+2=2 n(n+1)$, which means finding the smallest number greater than 1000 that, when divided by 3, leaves a remainder of 1, when divided by 5, leaves a remainder of 2, and can be expressed in the form $2 n(n+1)$. First, from $2 n(n+1)$, this number must be even and a multiple of 4. From $5 m+2$, the last digit of this number must be 2. Considering the smallest number, we examine 1002, 1012, 1022, 1032, .... Clearly, 1002 is a multiple of 3, leaving a remainder of 0 when divided by 3, so it is discarded. 1012 leaves a remainder of 1 when divided by 3, and $1012=2^{2} \times 11 \times 23=2 \times 22 \times 23$, which meets the requirements, so the smallest number is 1012.
Note: If 1012 cannot be expressed in the form $2 n(n+1)$, then the next number to consider is not 1022 or 1032. Instead, by the requirement of leaving a remainder of 1 when divided by 3, we directly consider $1012+30=1042$, and the next one is $1072 \ldots \ldots$ until a number that meets the problem's requirements is found.