Math Notes

Turn each of these palindromes “inside out” and their sum remains the same:

13031 + 42024 + 53035 + 57075 + 68086 + 97079 = 31013 + 24042 + 35053 + 75057 + 86068 + 79097

Remarkably, this holds true even if you square or cube them:

130312 + 420242 + 530352 + 570752 + 680862 + 970792 = 310132 + 240422 + 350532 + 750572 + 860682 + 790972

130313 + 420243 + 530353 + 570753 + 680863 + 970793 = 310133 + 240423 + 350533 + 750573 + 860683 + 790973

From Albert Beiler, Recreations in the Theory of Numbers, 1964.


  • Douglas Adams claimed that the funniest three-digit number is 359.
  • Romeo has more lines than Juliet, Iago than Othello, and Portia than Shylock.
  • Friday the 13th occurs at least once a year.
  • “By nature, men love newfangledness.” — Chaucer
  • John was the only apostle to die a natural death.

The Paradox of Dives and Lazarus

The Gospel of Luke contains a parable about a rich man and a beggar. Both men die, and the rich man is consigned to hell while the beggar is received into the bosom of Abraham. The rich man pleads for mercy, but Abraham tells him that in his lifetime he received good things and the beggar evil things: “now he is comforted and thou art tormented.” The rich man then begs that his brothers be warned of what lies in store for them, but Abraham rejects this plea as well, saying, “If they hear not Moses and the prophets, neither will they be persuaded though one rose from the dead.”

Now, writes E.V. Milner:

Suppose … that this last request of Dives had been granted; suppose, in fact, that some means were found to convince the living, whether rich men or beggars, that ‘justice would be done’ in a future life, then, it seems to me, an interesting paradox would emerge. For if I knew that the unhappiness which I suffer in this world would be recompensed by eternal bliss in the next world, then I should be happy in this world. But being happy in this world I should fail to qualify, so to speak, for happiness in the next world. Therefore, if there were such a recompense awaiting me, its existence would seem to entail that I should at least be not wholly convinced of its existence.

“Put epigrammatically, it would appear that the proposition ‘Justice will be done’ can only be true for one who believes it to be false. For one who believes it to be true justice is being done already.”


Write the numbers 82 to 1 in descending order and string them together:


The resulting 155-digit number is prime.

Benardete’s Book Paradox

Here is a book lying on a table. Open it. Look at the first page. Measure its thickness. It is very thick indeed for a single sheet of paper — one half inch thick. Now turn to the second page of the book. How thick is this second sheet of paper? One fourth inch thick. And the third page of the book, how thick is this third sheet of paper? One eighth inch thick, etc. ad infinitum. We are to posit not only that each page of the book is followed by an immediate successor the thickness of which is one half that of the immediately preceding page but also (and this is not unimportant) that each page is separated from page 1 by a finite number of pages. These two conditions are logically compatible: there is no certifiable contradiction in their joint assertion. But they mutually entail that there is no last page in the book. Close the book. Turn it over so that the front cover of the book is now lying face down upon the table. Now, slowly lift the back cover of the book with the aim of exposing to view the stack of pages lying beneath it. There is nothing to see. For there is no last page in the book to meet our gaze.

— Patrick Hughes and George Brecht, Vicious Circles and Infinity, 1978

The Unrepentant Liar

ushenko russell liar paradox

“According to [Bertrand] Russell’s treatment the sentence within the rectangle of Fig. 1 is meaningless, and may be called a pseudo-statement, because it is a version of the liar-paradox. But Russell’s treatment is unsatisfactory because it resolves the original paradox at the price of a new one. For, if the sentence of Fig. 1 is meaningless we must admit, since we observe that there are no other sentences within the rectangle, that it is false that there is a genuine or meaningful statement within the rectangle of Fig. 1. And, if there is no statement within the rectangle of Fig. 1 then it is false that there is a true statement within the rectangle of Fig. 1. The italicized part of the preceding sentence will be recognized as identical with (even if a different token of) the sentence within the rectangle of Fig. 1. And since the italicized sentence is true, and therefore a meaningful statement, the sentence within the rectangle is not a pseudo-statement either. Thus, if the sentence in question is meaningless, then it is meaningful and vice versa.”

— A.P. Ushenko, “A Note on the Liar Paradox,” Mind, October 1955

Faith No More

Kurt Gödel composed an ontological proof of God’s existence:

Axiom 1. A property is positive if and only if its negation is negative.

Axiom 2. A property is positive if it necessarily contains a positive property.

Theorem 1. A positive property is logically consistent (that is,
possibly it has an existence).

Definition. Something is God-like if and only if it possesses all positive properties.

Axiom 3. Being God-like is a positive property.

Axiom 4. Being a positive property is logical and hence necessary.

Definition. A property P is the essence of x if and only if x has the property P and P is necessarily minimal.

Theorem 2. If x is God-like, then being God-like is the essence of x.

Definition. x necessarily exists if it has an essential property.

Axiom 5. Being necessarily existent is God-like.

Theorem 3. Necessarily there is some x such that x is God-like.

“I am convinced of the afterlife, independent of theology,” he once wrote. “If the world is rationally constructed, there must be an afterlife.”