### Brain Teasers

# U.N.K.A.L

MADADIA has been chosen to host the latest United Nations Kebab Annual Liaisons (U.N.K.A.L).

Madadia's mayoress, Madka Owdiseez, has invited several wives of delegates of UNKAL for an informal luncheon.

She plans to seat her 9 guests in a row such that each lady will be able to converse with the person directly to her left and right. She has prepared the following list.

Mrs. F speaks English only.

Mrs. G speaks English and French.

Mrs. H speaks English and Russian.

Mrs. J speaks Russian only.

Mrs. K speaks English only.

Mrs. L speaks French only.

Mrs. M speaks French and German.

Mrs. N speaks English and German.

Mrs. O speaks English only.

How many distinct seating arrangements are possible? Give all possible seating arrangements. Bear in mind that Madka Owdiseez is not one of the 9 to be seated in a row.

Note that ABCD and DCBA are the same

Madadia's mayoress, Madka Owdiseez, has invited several wives of delegates of UNKAL for an informal luncheon.

She plans to seat her 9 guests in a row such that each lady will be able to converse with the person directly to her left and right. She has prepared the following list.

Mrs. F speaks English only.

Mrs. G speaks English and French.

Mrs. H speaks English and Russian.

Mrs. J speaks Russian only.

Mrs. K speaks English only.

Mrs. L speaks French only.

Mrs. M speaks French and German.

Mrs. N speaks English and German.

Mrs. O speaks English only.

How many distinct seating arrangements are possible? Give all possible seating arrangements. Bear in mind that Madka Owdiseez is not one of the 9 to be seated in a row.

Note that ABCD and DCBA are the same

### Answer

With each arrangement considered identical to its reverse, there are 126 possible arrangements.FKO are interchangeable, and can be arranged 6 ways. FKO stays together unless N is next to M outside GLM (more on that below).

JH must be together, and J must be at the end, because H is the only other Russian speaker. JH has 2 slots in the above configuration (H next to FKO, with J on the outside).

GLM has 1 slot in the above configuration (G next to FKO, opposite JH).

2[JH]*6[FKO]*1[MLG]=12 arrangements, or 2 mirror groups of 6 arrangements each. Thus, 6 distinct arrangements - 1 for each FKO order.

Keeping the above groups self-contained, N has 5 slots (2 within FKO, 1 between FKO and H, 1 between FKO and G, 1 beside M). That makes 30 arrangements - 5 for each FKO order.

When N is in certain positions, however, the groups can break up. Unless N is next to M outside GLM, GLM can be GML. If N is between FKO and GLM, GLM can scramble, and G can be anywhere touching FKO, except N cannot touch L. When G-[L/M]-N are consecutive in either direction, FKO can straddle that subgroup. That makes 126 arrangements - 21 for each FKO order.

Here are the 21 arrangements, using xxx for the interchangeable FKO.

JHNxxxGLM, JHNxxxGML, JHxNxxGLM, JHxNxxGML

JHxxNxGLM, JHxxNxGML, JHxxxNGLM, JHxxxNGML

JHxxxNMGL, JHxxxNMLG, JHxxNMLGx, JHxNMLGxx

JHNMLGxxx, JHxxxGLMN, JHxxGLMNx, JHxGLMNxx

JHGLMNxxx, JHGxxxNML, JHxGxxNML, JHxxGxNML

JHxxxGNML

Apply each FKO order - F-K-O, F-O-K, K-F-O, K-O-F, O-F-K, O-K-F - to each xxx above. That makes 21*6=126 arrangements.

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