Question 1

The sequence for an arrangement of stick figures tells us, for each figure, how many figures to its left are shorter than it.

Step 1 — Determine stick figure heights

Let us assign a unique height to each stick figure. We will use heights from 0 to 6. Let us look at sequence (c): 0, 1, 2, 3, 4, 5, 6. The arrangement for this sequence is FDBGCEA. For the first figure F, there are 0 shorter figures to its left. This is always true. For the second figure D, there is 1 shorter figure to its left. This means F is shorter than D. For the third figure B, there are 2 shorter figures to its left. This means F and D are shorter than B. This pattern continues for all figures. Each figure is taller than all figures placed before it. So, the heights must be in increasing order: F < D < B < G < C < E < A. Let us assign the heights: $H_F = 0$ $H_D = 1$ $H_B = 2$ $H_G = 3$ $H_C = 4$ $H_E = 5$ $H_A = 6$ Let us check this with sequence (b): 0, 0, 0, 0, 0, 0, 0. The arrangement for this sequence is AECGBDF. For the first figure A, there are 0 shorter figures to its left. For the second figure E, there are 0 shorter figures to its left. This means A is not shorter than E. So A is taller than E. This pattern continues for all figures. Each figure is shorter than all figures placed before it. So, the heights must be in decreasing order: A > E > C > G > B > D > F. This matches our assigned heights perfectly.

$\boxed{\text{Stick figure heights: F=0, D=1, B=2, G=3, C=4, E=5, A=6}}$

Step 2 — Verify arrangement for (a) 0, 1, 1, 2, 4, 1, 5

The required arrangement is FCBGADE. Let us list the figures in this order with their heights: F(0), C(4), B(2), G(3), A(6), D(1), E(5). Now, let us count shorter figures to the left for each position:

1. For F(0): No figures are to its left. So, 0 shorter figures.
2. For C(4): Figure to its left is F(0). F(0) is shorter than C(4). So, 1 shorter figure.
3. For B(2): Figures to its left are F(0), C(4). F(0) is shorter than B(2). C(4) is taller than B(2). So, 1 shorter figure.
4. For G(3): Figures to its left are F(0), C(4), B(2). F(0) and B(2) are shorter than G(3). C(4) is taller. So, 2 shorter figures.
5. For A(6): Figures to its left are F(0), C(4), B(2), G(3). All are shorter than A(6). So, 4 shorter figures.
6. For D(1): Figures to its left are F(0), C(4), B(2), G(3), A(6). F(0) is shorter than D(1). Others are taller. So, 1 shorter figure.
7. For E(5): Figures to its left are F(0), C(4), B(2), G(3), A(6), D(1). F(0), C(4), B(2), G(3), D(1) are shorter than E(5). A(6) is taller. So, 5 shorter figures. The sequence obtained is 0, 1, 1, 2, 4, 1, 5. This matches the given sequence.

$\boxed{\text{Arrangement for (a) is FCBGADE}}$

Step 3 — Verify arrangement for (b) 0, 0, 0, 0, 0, 0, 0

The required arrangement is AECGBDF. Let us list the figures in this order with their heights: A(6), E(5), C(4), G(3), B(2), D(1), F(0). Now, let us count shorter figures to the left for each position:

1. For A(6): No figures are to its left. So, 0 shorter figures.
2. For E(5): Figure to its left is A(6). A(6) is taller than E(5). So, 0 shorter figures.
3. For C(4): Figures to its left are A(6), E(5). Both are taller than C(4). So, 0 shorter figures.
4. For G(3): Figures to its left are A(6), E(5), C(4). All are taller than G(3). So, 0 shorter figures.
5. For B(2): Figures to its left are A(6), E(5), C(4), G(3). All are taller than B(2). So, 0 shorter figures.
6. For D(1): Figures to its left are A(6), E(5), C(4), G(3), B(2). All are taller than D(1). So, 0 shorter figures.
7. For F(0): Figures to its left are A(6), E(5), C(4), G(3), B(2), D(1). All are taller than F(0). So, 0 shorter figures. The sequence obtained is 0, 0, 0, 0, 0, 0, 0. This matches the given sequence.

$\boxed{\text{Arrangement for (b) is AECGBDF}}$

Step 4 — Verify arrangement for (c) 0, 1, 2, 3, 4, 5, 6

The required arrangement is FDBGCEA. Let us list the figures in this order with their heights: F(0), D(1), B(2), G(3), C(4), E(5), A(6). Now, let us count shorter figures to the left for each position:

1. For F(0): No figures are to its left. So, 0 shorter figures.
2. For D(1): Figure to its left is F(0). F(0) is shorter than D(1). So, 1 shorter figure.
3. For B(2): Figures to its left are F(0), D(1). Both are shorter than B(2). So, 2 shorter figures.
4. For G(3): Figures to its left are F(0), D(1), B(2). All are shorter than G(3). So, 3 shorter figures.
5. For C(4): Figures to its left are F(0), D(1), B(2), G(3). All are shorter than C(4). So, 4 shorter figures.
6. For E(5): Figures to its left are F(0), D(1), B(2), G(3), C(4). All are shorter than E(5). So, 5 shorter figures.
7. For A(6): Figures to its left are F(0), D(1), B(2), G(3), C(4), E(5). All are shorter than A(6). So, 6 shorter figures. The sequence obtained is 0, 1, 2, 3, 4, 5, 6. This matches the given sequence.

$\boxed{\text{Arrangement for (c) is FDBGCEA}}$

Step 5 — Verify arrangement for (d) 0, 1, 0, 1, 0, 1, 0

The required arrangement is EAGCDBF. Let us list the figures in this order with their heights: E(5), A(6), G(3), C(4), D(1), B(2), F(0). Now, let us count shorter figures to the left for each position:

1. For E(5): No figures are to its left. So, 0 shorter figures.
2. For A(6): Figure to its left is E(5). E(5) is shorter than A(6). So, 1 shorter figure.
3. For G(3): Figures to its left are E(5), A(6). Both are taller than G(3). So, 0 shorter figures.
4. For C(4): Figures to its left are E(5), A(6), G(3). G(3) is shorter than C(4). E(5) and A(6) are taller. So, 1 shorter figure.
5. For D(1): Figures to its left are E(5), A(6), G(3), C(4). All are taller than D(1). So, 0 shorter figures.
6. For B(2): Figures to its left are E(5), A(6), G(3), C(4), D(1). D(1) is shorter than B(2). Others are taller. So, 1 shorter figure.
7. For F(0): Figures to its left are E(5), A(6), G(3), C(4), D(1), B(2). All are taller than F(0). So, 0 shorter figures. The sequence obtained is 0, 1, 0, 1, 0, 1, 0. This matches the given sequence.

$\boxed{\text{Arrangement for (d) is EAGCDBF}}$

Step 6 — Verify arrangement for (e) 0, 1, 1, 1, 1, 1, 1

The required arrangement is FAECGBD. Let us list the figures in this order with their heights: F(0), A(6), E(5), C(4), G(3), B(2), D(1). Now, let us count shorter figures to the left for each position:

1. For F(0): No figures are to its left. So, 0 shorter figures.
2. For A(6): Figure to its left is F(0). F(0) is shorter than A(6). So, 1 shorter figure.
3. For E(5): Figures to its left are F(0), A(6). F(0) is shorter than E(5). A(6) is taller. So, 1 shorter figure.
4. For C(4): Figures to its left are F(0), A(6), E(5). F(0) is shorter than C(4). A(6) and E(5) are taller. So, 1 shorter figure.
5. For G(3): Figures to its left are F(0), A(6), E(5), C(4). F(0) is shorter than G(3). A(6), E(5), C(4) are taller. So, 1 shorter figure.
6. For B(2): Figures to its left are F(0), A(6), E(5), C(4), G(3). F(0) is shorter than B(2). Others are taller. So, 1 shorter figure.
7. For D(1): Figures to its left are F(0), A(6), E(5), C(4), G(3), B(2). F(0) is shorter than D(1). Others are taller. So, 1 shorter figure. The sequence obtained is 0, 1, 1, 1, 1, 1, 1. This matches the given sequence.

$\boxed{\text{Arrangement for (e) is FAECGBD}}$

Step 7 — Verify arrangement for (f) 0, 0, 0, 3, 3, 3, 3

The required arrangement is BDFAECG. Let us list the figures in this order with their heights: B(2), D(1), F(0), A(6), E(5), C(4), G(3). Now, let us count shorter figures to the left for each position:

1. For B(2): No figures are to its left. So, 0 shorter figures.
2. For D(1): Figure to its left is B(2). B(2) is taller than D(1). So, 0 shorter figures.
3. For F(0): Figures to its left are B(2), D(1). Both are taller than F(0). So, 0 shorter figures.
4. For A(6): Figures to its left are B(2), D(1), F(0). All are shorter than A(6). So, 3 shorter figures.
5. For E(5): Figures to its left are B(2), D(1), F(0), A(6). B(2), D(1), F(0) are shorter than E(5). A(6) is taller. So, 3 shorter figures.
6. For C(4): Figures to its left are B(2), D(1), F(0), A(6), E(5). B(2), D(1), F(0) are shorter than C(4). A(6) and E(5) are taller. So, 3 shorter figures.
7. For G(3): Figures to its left are B(2), D(1), F(0), A(6), E(5), C(4). B(2), D(1), F(0) are shorter than G(3). A(6), E(5), C(4) are taller. So, 3 shorter figures. The sequence obtained is 0, 0, 0, 3, 3, 3, 3. This matches the given sequence.

$\boxed{\text{Arrangement for (f) is BDFAECG}}$

Answer

(a) The required arrangement is FCBGADE. (b) The required arrangement is AECGBDF. (c) The required arrangement is FDBGCEA. (d) The required arrangement is EAGCDBF. (e) The required arrangement is FAECGBD. (f) The required arrangement is BDFAECG.