2020 AMC Help

Carnation

Anyone knows how to solve the following two questions?

1. Starting with a 9×9×9 cube, Augusta removed
as few 1 × 1 × 1 cubes as possible so that the
resulting sculpture had front view, top view and
side view all the same, as shown.
How many 1 × 1 × 1 cubes did Augusta remove?



2. My grandson makes wall hangings by stitching
together 16 square patches of fabric into a 4 × 4
grid. I asked him to use patches of red, blue,
green and yellow, but to ensure that no patch
touches another of the same colour, not even diagonally.
The picture shows an attempt which fails only
because two yellow patches touch diagonally.
In how many different ways can my grandson
choose to arrange the coloured patches correctly?

G B R Y
R Y G B
G B Y R
Y R G B

scott.songgps

Q28.
check the bigger hole first,It should cut through 7 3x3x3 cubes total 189
the rest 20 3x3x3 every one been cut by a small hole 7 of them each.total 20x7=140
two kinds of holes together 140+189=329

act_nan

28题是329，可以拆成27个3*3*3的小立方体来考虑。
30题是168，可以用数独的做法从上到下从左到右推算。

scott.songgps

Q30:
for every GBRY in different positions we have 24 ways to make it.(GBRY, GYRB...)
for GBRY , GBGY ... we have 7 different displays so,for total different ways we have 7x24=168 ways.

Carnation

scott.songgps 发表于 2021-4-13 22:48
Q30:
for every GBRY in different positions we have 24 ways to make it.(GBRY, GYRB...)
for GBRY , GBG ...

多谢，这个 for GBRY , GBGY ... we have 7 different displays我还是没明白，你可以讲更详细一点？

golfa

这种题挺难的，时间又紧

scott.songgps

Carnation 发表于 2021-4-14 15:23
多谢，这个 for GBRY , GBGY ... we have 7 different displays我还是没明白，你可以讲更详细一点？ ...

等有空，画个表给你。