This week's puzzle poses an evident paradox. Provided ten self-referential statements, your task is to recognize which that statements space true, and also which room false.

You are watching: Which of these statements is true?

### Sunday Puzzle #23: explanation of Truth

This week's puzzle is presented at the top of the post. I will repeat that here. Above, there room ten numbered statements. Which of them is/are true? What is your reasoning?

We'll be back next week with the equipment – and also a brand-new puzzle! got a good brainteaser, original or otherwise, the you'd like to watch featured? E-mail me with your recommendations. Together always, be sure to incorporate "Sunday Puzzle" in the subject line!

### Solution to Sunday Puzzle #22: Archimedes' cattle Problem

Last week, I challenged you to take on one of Archimedes' most daunting puzzles. As plenty of of you conveniently deduced, Archimedes' cattle Problem is, in fact, a mathematics problem. You can be of the exact same mind as commenter pyronius, who, top top recognizing the arithmetic nature of critical week's puzzle, remarked:

"Yo, archie. Disguising math as cows does no a puzzle make."

You and also pyronius would certainly be wrong. Numerous of the greatest puzzles space in truth math difficulties in disguise. That said, I know the frustration end this particular puzzle. Partly because the language that the problem itself deserve to be daunting to parse (this thread, initiated by commenter hawkingdo, walk a good job the translating the puzzle into some workable numbers); but likewise because Archimedes' cattle Problem – the second part, specifically – is a actual doozy. As with most multi-step problems, fixing it requires a careful, organized approach.

PART i SOLUTION

The very first person to provide not just a correct equipment to the very first part the the puzzle yet an account the his work was commenter zachparker, that wrote (solutions appear in bold):

First points first, ns converted every little thing to equations, favor so:

W = 5/6B + Y

B = 9/20D + Y

D = 13/42W + Y

w = 7/12B + 7/12b

b = 9/20D + 9/20d

d = 11/30Y + 11/30y

y = 13/42W + 13/42

This gives 7 equations because that 8 variables, so ns threw in an extra one:

W = 1

which would certainly yield all the other variables together a proportion of W.

See more: How Many Ounces Are In A Ton To Ounces, Convert 1 Ton To Ounces

Seeing as I really didn't feel choose doing every the algebra by hand, i recalculated every of the above equations to have a 0 come the appropriate of each =, and built a pair of matrices, one 8x8 for the left sides and an 8x1 because that the ride sides, and plugged the matrices into my trusty TI-89 (I tried it in Excel first, and also got a bunch that unusable decimals, and I want to keep my number nice and fractional in bespeak to find an LCD).

Running M1^-1 * M2 got me the adhering to results:

W= 1

B = 267/371

Y = 297/742

D = 790/1113

w = 171580/246821

b = 815541/1727747

y = 1813071/3455494

d = 83710/246821

Since ns knew each of this numbers were together a fraction of W, I discovered the LCD of all the fractions (1036648) and multiplied it by each fraction, to gain the following quantities:

W = 10,366,482

B = 7,460,514

Y = 4,149,387

D = 7,358,060

w = 7,206,360

b = 4,893,246

y = 5,439,213

d = 3,515,820

As zachparker notes, this problem lends chin to evaluation by what's known in direct algebra as a coefficient matrix. Such matrices are quickly resolved with a program, or, in zachparker's case, a fancy calculator. Yet in the absence of this tools, the problem's 7 equations and eight unknowns can additionally be addressed for by hand. Commenter IrkedIndeed did specifically that, and also came up v the exact same solution as zachparker. You have the right to read IrkedIndeed's technique here, but a cleaner presentation that the necessary steps is noted in David Wells' Book of Curious and also Interesting Puzzles: