Republic of Mathematics

Mathematics of the people, for the people, by the people: encouraging mathematical happiness.

[Republic of Mathematics @republicofmath]

Hi everyone - I've opened a BlueSky account: @republicofmath.bsky.social. See you there!

[Republic of Mathematics @republicofmath]

Geometric proofs of the irrationality of square-roots for select integers
https://t.co/avMXsq9nCD

[Republic of Mathematics @republicofmath]

@dmarain @jamestanton Whoops!!

[Republic of Mathematics @republicofmath]

@jamestanton @iconjack Is base 9 robust? 1/7 is the issue:
1/2 = 1/2 (base 9)
1/3 = 375/1246 (base 9)
1/4 = 1542/6378 (base 9)
1/5 = 756/4213 (base 9)
1/6 = 2/13 (base 9)
1/7 = ?? /???(base 9)
1/8 = 417/3652 (base 9)

[Justin Lanier @j_lanier]

Hi! Next month I'm running a learning experience for K-12 teachers called "21st Century Mathematics". Its goal is to help teachers to connect the math they teach with some math that has been discovered in the 21st century.

Info+register: https://t.co/w0STBSwNm1

Pass it along!

[Republic of Mathematics @republicofmath]

@jamestanton @iconjack Base 8 is robust:
1/2 = 1/2 (base 8)
1/3 = 347/1265 (base 8)
1/4 = 275/1364 (base 8)
1/5 = 653/4127 (base 8)
1/6 = 245/1736 (base 8)
1/7 = 421/3567 (base 8)

[Republic of Mathematics @republicofmath]

@jamestanton @iconjack Is base 7 robust:
1/2 = 1/2 (base 7)
1/3 = 3/12 (base 7)
1/4 = 54/312 (base 7)
1/5 = 2/13 (base 7)
1/6 = ?? ?

[Republic of Mathematics @republicofmath]

@jamestanton @iconjack 1/3 seems to be problematic bases 5 and 6, but 1/3 = 3/12 (base 7)

[Republic of Mathematics @republicofmath]

@jamestanton @iconjack 1/5 = 41/325 (base 6)

[Republic of Mathematics @republicofmath]

@jamestanton @iconjack 1/4 = 2/13 (base 5) but 1/3 = ??

[Republic of Mathematics @republicofmath]

@jamestanton @iconjack Statement for 1/6 should be 1/6 = 571/3426

[Republic of Mathematics @republicofmath]

@dmarain @jamestanton 1/6 = 571/3426 and 1/7 = 3/21

[James Tanton @jamestanton]

Base b is "robust" if each fraction 1/2,1/3,..,1/(b-1) is a ratio of two integers expressed in base b that use each digit 1, 2, 3,.... (up to some value) exactly once. (1/b cannot be so expressed. Why?)
@iconjack shows base 10 is robust.
Base 3 and base 4 are robust. Base 5? 6?

[Republic of Mathematics @republicofmath]

Dominic Welsh's important contributions to probability and combinatorics: https://t.co/ltL80pg1Ty

[Republic of Mathematics @republicofmath]

@jamestanton Express each 1/n, n= 1, ..., 9 in the simplest form as a ratio of two integers where each of the digits 1, 2, 3, .... (up to some value) are used exactly once.
What about 1/n for n>= 10?

[Republic of Mathematics @republicofmath]

@jamestanton And 1/9 = 6381/57429

[Republic of Mathematics @republicofmath]

@jamestanton Yes: 1/3 = 1746/5238

[Republic of Mathematics @republicofmath]

Data from 350,757 coin flips provide strong evidence that when some (but not all) people flip a fair coin, it tends to land on the same side it started.
https://t.co/zSljdyeLIR

[James Tanton @jamestanton]

N>1 cups all upright. A "move" consists of turning all but 2 cups over. (So, each move N-2 cups change state upright/upside-down.)
For which N is it possible to make all N cups upside-down?
[General theory about N cups and turning N-k over at a time?]