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An ant is at a crossroads of a square grid of highways spaced 1 meter apart.
The ant can walk 1 meter in 15 seconds along a highway, but off-road it can only travel half as fast.
Find the area of the region composed of all points the ant can reach in at most 30 seconds.
Note: an ant highway is just a line with no thickness.
476I2 P54274Y 102 38918 02 "3I8G 9F 59C3
& 5977 "
How does the above sentence relate
to to-day's date?
Finding the only zero-less answer to the alphametic
might be of help.
Graph y = x3
/3 - x2
Imagine a small circle sitting inside the local minimum. Gravity pulls in the negative y direction so it is quite stable there.
Now increase the size of the circle. At some point it will become too large to fit in this hollow and will be forced to roll away down into the third quadrant.
At what radius does this happen?
I have a square table that used to be perfectly stable, back when it had a foot at the end of each leg. Now the feet are missing from legs B, C, and D There's only one left at leg A.
The table measures 50 cm x 50 cm with legs 99.5 cm long. The foot is flat and attached by a swivel so that the end of leg A is a constant 0.5 cm from the ground. (In other words, the total length of leg A plus the foot is 100cm, 0.5 cm longer than each of the other three.)
The table can now rock back and forth with the foot of A and its opposite leg C in constant contact with the floor.
If B is also in contact with the floor, how far is D from the floor?
Note: consider the tips of the legs to be singular points at the corners of a square.
Express 31 by 4 successive digits, using no more than 4 out of 5 basic operations (i.e. + - / * and exponentation), nothing else allowed.
The order of the digits must be an arithmetic progression like 8*7-6-5 or 3+4*5+6, examples fitting the specifications but not the result.
The (missing adjective) number X
is the product of three or four or five consecutive numbers.
What is X ?
What is the missing adjective?
Five Consecutive Primes starting from M1 create two palindromic numbers:
M1*M2*...*M5 = Pal1
Find M1, Pal1 & Pal2.
I know only one solution...
A certain school rewards students for superior attendance and punctuality. The prize is received if there is no more than one lateness and no absence has lasted more than two consecutive days. If a student is late more than once in the specified duration of the award period or is absent three or more days in a row, the student is not eligible for a prize.
Note that multiple latenesses need not be on consecutive days to disentitle an award, but the three disqualifying absences need to be successive to cancel the award.
To keep track of attendance and punctuality for one student, a string of letters is used. For example:
PPPAAPAPPPAALAPPPPAP represents a sequence of Punctual days, Absence days and Lateness days that does qualify for the award, as none of the A's come in a group of three and there is only one lateness.
PPPAAPAPPPAAAPPPPAPL disqualifies the student because of the three absences in a row.
PPPAAPAPPPAALAPPPPAL disqualifies based on two latenesses.
A recent such prize period lasted 12 days. How many strings of 12 are there that would qualify for an award? There are obviously 3^12 possible strings altogether, many of which would not qualify; but how many do qualify?
Adapted from a Project Euler puzzle on-line.
This simple math problem went viral (about five million views) on YouTube:
What is your answer and its justification?
How far can you march on? e.g. mart, heart,....counterpart etc
Proper names allowed - any word should be supported by a respectable dictionary.
Bonus: try the same "march" with another 3-letter combination.
Katagiri calls a number nude if it is divisible by each of its digits
e.g like 672.
Clearly this definition (there exist other as well!) implies non-zero numbers only.
a. How many nude numbers are there below 1,000,000?
b. Evaluate the smallest nude number which contains all the odd digits.
c. Find the smallest triplet of consecutive nontrivial nude numbers.
d. Prove that there is infinite number of nude numbers.
Prove that every other Fibonacci number from 5 on (i.e. 5,13,34,89... etc) can denote the length of a hypotenuse of a right triangle with integer sides.
Find a sequence of 7 consecutive primes,
creating an arithmetic progression.
There are several solutions, all are big numbers.