The Banner of St. George - Futility Closet

A “simple but pretty little puzzle” by Henry Dudeney: If this flag measures 4 feet by 3 feet and presents equal areas of red and white bunting, how wide are the cross’s arms?

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The flag measures 4 feet by 3 feet, so its diagonal is 5 feet. “All you need to do is to deduct half the length of this diagonal (2 1/2 feet) from a quarter of the distance all round the edge of the flag (3 1/2 feet) — a quarter of 14 feet. The difference (1 ft.) is the required width of the arm of the red cross.” Proof: If the flag measures 4 feet by 3 feet, the red area is given by 4x + 3x – x² = 6, so x² – 7x + 6 = 0. Solving the quadratic equation gives x – 3.5 = ± 2.5, and as x can’t be 6, the answer must be 1 foot. From Dudeney’s Amusements in Mathematics, via Erwin Brecher’s Ultimate Book of Puzzles, Mathematical Diversions, and Brainteasers (1996). 04/18/2025 UPDATE: Reader Catalin Voinescu has a simpler solution: “Consider: the banner has the same areas of red and white as a banner of the same size with a red L of the same thickness as the cross along two of the edges. (Cut the original banner in four and rearrange the fragments.) In this case, the white area is a rectangle. The problem states that it must be half of the area of the flag, that is, 6 sq ft. Its sides are shorter than the sides of the banner by the same amount each, namely, the thickness of the red L. Without any equation, one immediately notices that 3 by 2 gives 6, and that’s 1 less than 4 by 3. So the red part is 1 ft wide.” (Thanks, Catalin.)

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The flag measures 4 feet by 3 feet, so its diagonal is 5 feet. “All you need to do is to deduct half the length of this diagonal (2 1/2 feet) from a quarter of the distance all round the edge of the flag (3 1/2 feet) — a quarter of 14 feet. The difference (1 ft.) is the required width of the arm of the red cross.”

Proof:

If the flag measures 4 feet by 3 feet, the red area is given by

4x + 3x – x² = 6,

x² – 7x + 6 = 0.

Solving the quadratic equation gives x – 3.5 = ± 2.5, and as x can’t be 6, the answer must be 1 foot.

From Dudeney’s Amusements in Mathematics, via Erwin Brecher’s Ultimate Book of Puzzles, Mathematical Diversions, and Brainteasers (1996).

04/18/2025 UPDATE: Reader Catalin Voinescu has a simpler solution:

“Consider: the banner has the same areas of red and white as a banner of the same size with a red L of the same thickness as the cross along two of the edges. (Cut the original banner in four and rearrange the fragments.) In this case, the white area is a rectangle. The problem states that it must be half of the area of the flag, that is, 6 sq ft. Its sides are shorter than the sides of the banner by the same amount each, namely, the thickness of the red L. Without any equation, one immediately notices that 3 by 2 gives 6, and that’s 1 less than 4 by 3. So the red part is 1 ft wide.”

(Thanks, Catalin.)